{"version":"1.0.0","segments":[{"startTime":0.491,"endTime":3.658,"body":"(electronic swishing)"},{"startTime":5.16,"endTime":9.66,"body":"- This lecture looks at games"},{"startTime":5.16,"endTime":9.66,"body":"of chance like lotteries,"},{"startTime":9.66,"endTime":12.15,"body":"dice, even tossing coins."},{"startTime":12.15,"endTime":14.193,"body":"Can mathematics help us win?"},{"startTime":15.03,"endTime":17.97,"body":"We'll see how a Frenchman's"},{"startTime":15.03,"endTime":17.97,"body":"gambling problem"},{"startTime":17.97,"endTime":20.493,"body":"instigated the whole field of probability."},{"startTime":21.33,"endTime":23.07,"body":"We'll find out why you should never buy"},{"startTime":23.07,"endTime":25.871,"body":"a British lottery ticket on a Monday."},{"startTime":25.871,"endTime":30.871,"body":"And we'll learn how Voltaire"},{"startTime":25.871,"endTime":30.871,"body":"got rich on a lottery"},{"startTime":31.11,"endTime":33.273,"body":"using a simple bit of mathematics."},{"startTime":34.23,"endTime":39.23,"body":"But let's start with one of"},{"startTime":34.23,"endTime":39.23,"body":"the oldest games of chance,"},{"startTime":40.47,"endTime":41.523,"body":"and that's dice."},{"startTime":42.45,"endTime":44.43,"body":"So dice have many advantages."},{"startTime":44.43,"endTime":48.54,"body":"They're inexpensive, cheap"},{"startTime":44.43,"endTime":48.54,"body":"to make, easy to carry."},{"startTime":48.54,"endTime":49.56,"body":"You can put them in your pocket,"},{"startTime":49.56,"endTime":51.9,"body":"you can take them with"},{"startTime":49.56,"endTime":51.9,"body":"you wherever you go."},{"startTime":51.9,"endTime":54.12,"body":"And people have been"},{"startTime":51.9,"endTime":54.12,"body":"playing games with dice"},{"startTime":54.12,"endTime":56.13,"body":"for a long, long time."},{"startTime":56.13,"endTime":58.53,"body":"I want to tell you about a"},{"startTime":56.13,"endTime":58.53,"body":"game that was quite popular"},{"startTime":58.53,"endTime":60.75,"body":"in France a few centuries ago,"},{"startTime":60.75,"endTime":63.66,"body":"and I'll just say the dice I'm"},{"startTime":60.75,"endTime":63.66,"body":"going to be talking about today"},{"startTime":63.66,"endTime":65.61,"body":"are always the standard dice."},{"startTime":65.61,"endTime":67.89,"body":"So each die is a cube."},{"startTime":67.89,"endTime":70.95,"body":"It's got the numbers"},{"startTime":67.89,"endTime":70.95,"body":"one to six on its faces,"},{"startTime":70.95,"endTime":74.52,"body":"and we'll assume that they"},{"startTime":70.95,"endTime":74.52,"body":"are fair, they're unbiased."},{"startTime":74.52,"endTime":78.37,"body":"Chance of throwing any one"},{"startTime":74.52,"endTime":78.37,"body":"of the numbers is 1/6th."},{"startTime":78.37,"endTime":81.937,"body":"Okay, so your mate, Antoine,"},{"startTime":78.37,"endTime":81.937,"body":"comes up to you and says,"},{"startTime":81.937,"endTime":85.807,"body":"\"I bet you five francs"},{"startTime":81.937,"endTime":85.807,"body":"that I can throw a six"},{"startTime":85.807,"endTime":90.12,"body":"\"if you give me four goes,\""},{"startTime":85.807,"endTime":90.12,"body":"should you take that bet?"},{"startTime":90.12,"endTime":91.74,"body":"What do we think?"},{"startTime":91.74,"endTime":94.563,"body":"So let me tell you a little"},{"startTime":91.74,"endTime":94.563,"body":"bit more about Antoine."},{"startTime":95.46,"endTime":99.99,"body":"Full name, Antoine Gombaud"},{"startTime":95.46,"endTime":99.99,"body":"Chevalier de Mere, Nobleman."},{"startTime":99.99,"endTime":102.578,"body":"Well, sort of; he actually"},{"startTime":99.99,"endTime":102.578,"body":"gave himself that nickname,"},{"startTime":102.578,"endTime":106.29,"body":"(laughs) but everyone just"},{"startTime":102.578,"endTime":106.29,"body":"started to call him that anyway."},{"startTime":106.29,"endTime":108.66,"body":"And he was very interested in gambling"},{"startTime":108.66,"endTime":111.63,"body":"and in calculating mathematical odds."},{"startTime":111.63,"endTime":115.29,"body":"He got a bit stuck on"},{"startTime":111.63,"endTime":115.29,"body":"some of his calculations,"},{"startTime":115.29,"endTime":117.87,"body":"and one of them was related to this game,"},{"startTime":117.87,"endTime":119.01,"body":"or a version of it."},{"startTime":119.01,"endTime":121.92,"body":"So let's see what we can"},{"startTime":119.01,"endTime":121.92,"body":"find out about this game."},{"startTime":121.92,"endTime":124.98,"body":"So you've got four goes to throw a six,"},{"startTime":124.98,"endTime":128.07,"body":"and that means there are"},{"startTime":124.98,"endTime":128.07,"body":"actually five possible ways"},{"startTime":128.07,"endTime":130.89,"body":"the game can turn out; here they are."},{"startTime":130.89,"endTime":134.55,"body":"So you throw a six straight"},{"startTime":130.89,"endTime":134.55,"body":"away possibly, or you fail,"},{"startTime":134.55,"endTime":136.62,"body":"but then you manage on your second attempt"},{"startTime":136.62,"endTime":139.331,"body":"or your third attempt,"},{"startTime":136.62,"endTime":139.331,"body":"or your fourth attempt."},{"startTime":139.331,"endTime":142.02,"body":"And the only way to lose in this game"},{"startTime":142.02,"endTime":144.6,"body":"is to fail consecutively four times."},{"startTime":144.6,"endTime":146.917,"body":"So, of course, we're not going to say,"},{"startTime":146.917,"endTime":150.263,"body":"\"Well, that means you have"},{"startTime":146.917,"endTime":150.263,"body":"an 80% chance of winning,\""},{"startTime":150.263,"endTime":152.22,"body":"(laughs) 'cause four of the five outcomes"},{"startTime":152.22,"endTime":153.45,"body":"leads to you winning."},{"startTime":153.45,"endTime":154.283,"body":"We won't say that."},{"startTime":154.283,"endTime":156.02,"body":"And the reason we don't say that"},{"startTime":156.02,"endTime":158.25,"body":"is because we have no reason to think"},{"startTime":158.25,"endTime":162.15,"body":"that these outcomes"},{"startTime":158.25,"endTime":162.15,"body":"are all equally likely."},{"startTime":162.15,"endTime":165.84,"body":"So we need to understand"},{"startTime":162.15,"endTime":165.84,"body":"what are the probabilities"},{"startTime":165.84,"endTime":167.82,"body":"of each of these possible outcomes."},{"startTime":167.82,"endTime":170.37,"body":"Okay, well, the first one,"},{"startTime":167.82,"endTime":170.37,"body":"maybe you just throw a six;"},{"startTime":170.37,"endTime":172.5,"body":"you're lucky straightaway,"},{"startTime":170.37,"endTime":172.5,"body":"you throw a six."},{"startTime":172.5,"endTime":174.967,"body":"The chance of that"},{"startTime":172.5,"endTime":174.967,"body":"happening is one in six."},{"startTime":174.967,"endTime":179.4,"body":"There are six numbers on the"},{"startTime":174.967,"endTime":179.4,"body":"die, and six is one of them."},{"startTime":179.4,"endTime":181.89,"body":"There's a one in six"},{"startTime":179.4,"endTime":181.89,"body":"chance of that happening."},{"startTime":181.89,"endTime":183.69,"body":"Fine; what about the next one?"},{"startTime":183.69,"endTime":187.44,"body":"Well, in that outcome, we"},{"startTime":183.69,"endTime":187.44,"body":"failed on our first attempt,"},{"startTime":187.44,"endTime":189.057,"body":"and then we are lucky"},{"startTime":187.44,"endTime":189.057,"body":"and we do throw a six."},{"startTime":189.057,"endTime":193.86,"body":"Now, the probability of"},{"startTime":189.057,"endTime":193.86,"body":"throwing a six is one in six."},{"startTime":193.86,"endTime":196.8,"body":"So the probability of throwing a not-six,"},{"startTime":196.8,"endTime":200.28,"body":"one of the other five"},{"startTime":196.8,"endTime":200.28,"body":"numbers, will be 5/6."},{"startTime":200.28,"endTime":202.8,"body":"So I claim that the"},{"startTime":200.28,"endTime":202.8,"body":"chance of throwing non-six"},{"startTime":202.8,"endTime":207.12,"body":"followed by six will be 5/6 times 1/6,"},{"startTime":207.12,"endTime":209.85,"body":"which is five out of 36."},{"startTime":209.85,"endTime":211.8,"body":"Now, why am I saying that?"},{"startTime":211.8,"endTime":215.4,"body":"Well, it's an example of a"},{"startTime":211.8,"endTime":215.4,"body":"general rule of probability"},{"startTime":215.4,"endTime":218.4,"body":"that says if you're looking"},{"startTime":215.4,"endTime":218.4,"body":"at independent events,"},{"startTime":218.4,"endTime":221.28,"body":"ones which do not affect each other,"},{"startTime":221.28,"endTime":224.61,"body":"the outcome of one does not"},{"startTime":221.28,"endTime":224.61,"body":"affect the outcome of the next,"},{"startTime":224.61,"endTime":227.1,"body":"then there's a rule that says this,"},{"startTime":227.1,"endTime":231.15,"body":"that if the probability"},{"startTime":227.1,"endTime":231.15,"body":"of event A is P, say,"},{"startTime":231.15,"endTime":234.06,"body":"and the probability of event B is Q, say,"},{"startTime":234.06,"endTime":236.13,"body":"then if they are"},{"startTime":234.06,"endTime":236.13,"body":"independent of each other,"},{"startTime":236.13,"endTime":237.99,"body":"they don't affect each other,"},{"startTime":237.99,"endTime":240.15,"body":"like consecutive rolls of a dice,"},{"startTime":240.15,"endTime":242.94,"body":"then the probability of both"},{"startTime":240.15,"endTime":242.94,"body":"of those things happening,"},{"startTime":242.94,"endTime":245.46,"body":"you just multiply together P times Q."},{"startTime":245.46,"endTime":249.0,"body":"Now, I must stress they have"},{"startTime":245.46,"endTime":249.0,"body":"to be independent events."},{"startTime":249.0,"endTime":252.39,"body":"They cannot be events that"},{"startTime":249.0,"endTime":252.39,"body":"influence each other in any way."},{"startTime":252.39,"endTime":255.48,"body":"For example, here is a false conclusion,"},{"startTime":255.48,"endTime":257.55,"body":"one that you should not apply this rule."},{"startTime":257.55,"endTime":262.55,"body":"If we know, say, that about"},{"startTime":257.55,"endTime":262.55,"body":"1/10th of adults have a beard,"},{"startTime":262.65,"endTime":266.46,"body":"and we know that about"},{"startTime":262.65,"endTime":266.46,"body":"one-half of adults are women,"},{"startTime":266.46,"endTime":268.41,"body":"we cannot multiply those together"},{"startTime":268.41,"endTime":272.97,"body":"and conclude that one adult"},{"startTime":268.41,"endTime":272.97,"body":"in 20 is a woman with a beard."},{"startTime":272.97,"endTime":276.9,"body":"Not true, however"},{"startTime":272.97,"endTime":276.9,"body":"glorious that would look."},{"startTime":276.9,"endTime":279.87,"body":"So however, luckily for us,"},{"startTime":279.87,"endTime":283.46,"body":"throws of a die are"},{"startTime":279.87,"endTime":283.46,"body":"independent of each other."},{"startTime":283.46,"endTime":285.03,"body":"So the outcome of one roll"},{"startTime":285.03,"endTime":286.74,"body":"does not affect the outcome of the next."},{"startTime":286.74,"endTime":289.89,"body":"So we really can multiply"},{"startTime":286.74,"endTime":289.89,"body":"those probabilities together"},{"startTime":289.89,"endTime":293.49,"body":"and get, as we've seen, five out of 36,"},{"startTime":293.49,"endTime":297.42,"body":"a probability of five in 36"},{"startTime":293.49,"endTime":297.42,"body":"of the second outcome there."},{"startTime":297.42,"endTime":299.4,"body":"And you can repeat that idea"},{"startTime":299.4,"endTime":301.89,"body":"and work out the chance of having non-six"},{"startTime":301.89,"endTime":304.263,"body":"followed by non-six, followed by six,"},{"startTime":305.16,"endTime":307.26,"body":"and non-non-non-six, and then finally,"},{"startTime":307.26,"endTime":308.91,"body":"a six on your fourth attempt."},{"startTime":308.91,"endTime":312.78,"body":"And you can get all of these"},{"startTime":308.91,"endTime":312.78,"body":"written down, there they are,"},{"startTime":312.78,"endTime":313.92,"body":"those are all those probabilities."},{"startTime":313.92,"endTime":315.33,"body":"You can do the calculations."},{"startTime":315.33,"endTime":317.64,"body":"So what's the chance"},{"startTime":315.33,"endTime":317.64,"body":"of winning in this game"},{"startTime":317.64,"endTime":319.8,"body":"of eventually managing to get a six"},{"startTime":319.8,"endTime":321.453,"body":"if you have four goes at it?"},{"startTime":322.41,"endTime":325.23,"body":"Well, we can just add up"},{"startTime":322.41,"endTime":325.23,"body":"all of those four things,"},{"startTime":325.23,"endTime":327.51,"body":"and if you do that, as"},{"startTime":325.23,"endTime":327.51,"body":"I'm sure you have all"},{"startTime":327.51,"endTime":332.243,"body":"just done in your head,"},{"startTime":327.51,"endTime":332.243,"body":"you get 671 out of 1,296,"},{"startTime":333.93,"endTime":338.67,"body":"which is about a 52% chance"},{"startTime":333.93,"endTime":338.67,"body":"of winning at this game."},{"startTime":338.67,"endTime":342.39,"body":"So we should not take"},{"startTime":338.67,"endTime":342.39,"body":"our friend Antoine's bet"},{"startTime":342.39,"endTime":345.57,"body":"when he sidled up to us,"},{"startTime":342.39,"endTime":345.57,"body":"because over the long-term,"},{"startTime":345.57,"endTime":350.28,"body":"he will have a better chance"},{"startTime":345.57,"endTime":350.28,"body":"of throwing that six than not."},{"startTime":350.28,"endTime":353.07,"body":"Now, there's actually a better"},{"startTime":350.28,"endTime":353.07,"body":"way to do this calculation."},{"startTime":353.07,"endTime":355.05,"body":"We worked out those"},{"startTime":353.07,"endTime":355.05,"body":"four things step-by-step"},{"startTime":355.05,"endTime":356.16,"body":"and adding them all up."},{"startTime":356.16,"endTime":358.057,"body":"But instead, we could just say,"},{"startTime":358.057,"endTime":360.457,"body":"\"Well, our chance of winning"},{"startTime":360.457,"endTime":362.697,"body":"\"is one minus our chance of losing.\""},{"startTime":363.57,"endTime":366.81,"body":"So if we have a, well, we can"},{"startTime":363.57,"endTime":366.81,"body":"work out the chance of losing."},{"startTime":366.81,"endTime":370.59,"body":"The only way to lose is to"},{"startTime":366.81,"endTime":370.59,"body":"fail four consecutive times"},{"startTime":370.59,"endTime":372.12,"body":"to throw a six."},{"startTime":372.12,"endTime":376.38,"body":"So that's a 5/6 chance,"},{"startTime":372.12,"endTime":376.38,"body":"multiplied by itself four times."},{"startTime":376.38,"endTime":378.33,"body":"We're trying it four times, okay?"},{"startTime":378.33,"endTime":382.47,"body":"So we get this calculation"},{"startTime":378.33,"endTime":382.47,"body":"here, 5/6 the power of four,"},{"startTime":382.47,"endTime":384.72,"body":"625 over 1,296."},{"startTime":384.72,"endTime":386.588,"body":"That is the chance of losing."},{"startTime":386.588,"endTime":388.23,"body":"It's the only way we can lose."},{"startTime":388.23,"endTime":389.85,"body":"So if that's the chance of losing,"},{"startTime":389.85,"endTime":392.91,"body":"then the chance of"},{"startTime":389.85,"endTime":392.91,"body":"winning is one minus that,"},{"startTime":392.91,"endTime":394.59,"body":"and we get that same outcome at the end."},{"startTime":394.59,"endTime":398.757,"body":"So that's a slightly quicker"},{"startTime":394.59,"endTime":398.757,"body":"way if we notice to do that."},{"startTime":398.757,"endTime":401.19,"body":"And that's going to be useful to us later."},{"startTime":401.19,"endTime":405.3,"body":"So after awhile, Antoine finds"},{"startTime":405.3,"endTime":407.16,"body":"that he's sort of getting"},{"startTime":405.3,"endTime":407.16,"body":"a bit bored of this game,"},{"startTime":407.16,"endTime":409.38,"body":"or maybe no one wants to"},{"startTime":407.16,"endTime":409.38,"body":"play it anymore with him"},{"startTime":409.38,"endTime":411.3,"body":"'cause he somehow keeps winning,"},{"startTime":411.3,"endTime":414.15,"body":"so he's experimenting with variants of it."},{"startTime":414.15,"endTime":415.53,"body":"And one he comes up with is this."},{"startTime":415.53,"endTime":418.95,"body":"Okay, we've been trying"},{"startTime":415.53,"endTime":418.95,"body":"to throw a single die"},{"startTime":418.95,"endTime":422.1,"body":"and throw a six, and we're"},{"startTime":418.95,"endTime":422.1,"body":"having four attempts at it."},{"startTime":422.1,"endTime":424.47,"body":"How about I make it a"},{"startTime":422.1,"endTime":424.47,"body":"little bit more spicy"},{"startTime":424.47,"endTime":427.53,"body":"and instead, I throw a"},{"startTime":424.47,"endTime":427.53,"body":"pair of dice, two dice,"},{"startTime":427.53,"endTime":429.93,"body":"and I try to get a double six."},{"startTime":429.93,"endTime":432.27,"body":"Now, the chances of getting a double six"},{"startTime":432.27,"endTime":435.417,"body":"while both of the dice have to be the six,"},{"startTime":435.417,"endTime":437.64,"body":"and there's a 1/6th"},{"startTime":435.417,"endTime":437.64,"body":"chance for each of them,"},{"startTime":437.64,"endTime":440.94,"body":"so 1/6 times 1/6th, one in 36."},{"startTime":440.94,"endTime":444.6,"body":"You've got a one in 36"},{"startTime":440.94,"endTime":444.6,"body":"chance on any given try"},{"startTime":444.6,"endTime":446.55,"body":"to throw a double six."},{"startTime":446.55,"endTime":451.53,"body":"So Gombaud said right, one"},{"startTime":446.55,"endTime":451.53,"body":"in 36, that's 1/6th as likely"},{"startTime":451.53,"endTime":454.2,"body":"as throwing a six with a single die."},{"startTime":454.2,"endTime":457.35,"body":"So there's this old"},{"startTime":454.2,"endTime":457.35,"body":"gambling rule that says"},{"startTime":457.35,"endTime":460.395,"body":"that what I can do is I"},{"startTime":457.35,"endTime":460.395,"body":"can say, well, this thing,"},{"startTime":460.395,"endTime":463.35,"body":"this alternative, is 1/6th as likely"},{"startTime":463.35,"endTime":465.63,"body":"as throwing a six with a single die."},{"startTime":465.63,"endTime":470.63,"body":"I know that that has a 52%"},{"startTime":465.63,"endTime":470.63,"body":"success rate with four attempts."},{"startTime":471.09,"endTime":474.54,"body":"So with this thing that's 1/6th"},{"startTime":471.09,"endTime":474.54,"body":"as likely what I should do"},{"startTime":474.54,"endTime":477.84,"body":"is have not four attempts"},{"startTime":474.54,"endTime":477.84,"body":"but four times six,"},{"startTime":477.84,"endTime":481.2,"body":"so you sort of make up"},{"startTime":477.84,"endTime":481.2,"body":"for the less likeliness,"},{"startTime":481.2,"endTime":483.84,"body":"you should have 24 attempts,"},{"startTime":481.2,"endTime":483.84,"body":"and that will give you"},{"startTime":483.84,"endTime":486.78,"body":"the same 52% likelihood of success."},{"startTime":486.78,"endTime":490.47,"body":"This is Gombaud's idea, and right,"},{"startTime":490.47,"endTime":492.18,"body":"so he thinks this is what it should be,"},{"startTime":492.18,"endTime":495.63,"body":"but unfortunately doesn't seem"},{"startTime":492.18,"endTime":495.63,"body":"to be working out that way."},{"startTime":495.63,"endTime":499.86,"body":"So he had a friend, and he"},{"startTime":495.63,"endTime":499.86,"body":"asked his friend for advice."},{"startTime":499.86,"endTime":501.36,"body":"And luckily for Gombaud,"},{"startTime":501.36,"endTime":504.39,"body":"his friend was one of the"},{"startTime":501.36,"endTime":504.39,"body":"best mathematicians in France,"},{"startTime":504.39,"endTime":508.625,"body":"Blaise Pascal, and Pascal,"},{"startTime":504.39,"endTime":508.625,"body":"he had a friend as well."},{"startTime":508.625,"endTime":511.53,"body":"And he asked his friend"},{"startTime":508.625,"endTime":511.53,"body":"about this question"},{"startTime":511.53,"endTime":513.06,"body":"and a couple of others, and his friend"},{"startTime":513.06,"endTime":515.73,"body":"was one of the other best"},{"startTime":513.06,"endTime":515.73,"body":"mathematicians in France,"},{"startTime":515.73,"endTime":516.87,"body":"Pierre de Fermat."},{"startTime":516.87,"endTime":518.484,"body":"Right, there we are, Pascal on the left."},{"startTime":518.484,"endTime":520.77,"body":"He's looking at cycloids in that statue,"},{"startTime":520.77,"endTime":523.26,"body":"it's a good statue; and"},{"startTime":520.77,"endTime":523.26,"body":"Fermat is on the right."},{"startTime":523.26,"endTime":525.897,"body":"So they have a series of"},{"startTime":523.26,"endTime":525.897,"body":"letters that go between them."},{"startTime":525.897,"endTime":527.34,"body":"And in that series of letters,"},{"startTime":527.34,"endTime":530.49,"body":"they pretty much establish"},{"startTime":527.34,"endTime":530.49,"body":"the field of probability"},{"startTime":530.49,"endTime":532.47,"body":"and mathematical study of probability."},{"startTime":532.47,"endTime":534.3,"body":"They get a lot of the basics right,"},{"startTime":534.3,"endTime":536.05,"body":"and established and worked-through."},{"startTime":537.0,"endTime":540.84,"body":"So we can solve this"},{"startTime":537.0,"endTime":540.84,"body":"question, was Gombaud right?"},{"startTime":540.84,"endTime":541.95,"body":"Was this intuition right?"},{"startTime":541.95,"endTime":544.35,"body":"And if not, what's the right answer?"},{"startTime":544.35,"endTime":547.268,"body":"So we're interested in"},{"startTime":544.35,"endTime":547.268,"body":"throwing a double six,"},{"startTime":547.268,"endTime":549.78,"body":"and we're going to have 24 goes at it."},{"startTime":549.78,"endTime":554.73,"body":"And Gombaud says this should"},{"startTime":549.78,"endTime":554.73,"body":"give you a 52% success rate,"},{"startTime":554.73,"endTime":557.31,"body":"so it should be a good"},{"startTime":554.73,"endTime":557.31,"body":"bet in the long run."},{"startTime":557.31,"endTime":560.67,"body":"Now, we don't want to add up 24"},{"startTime":557.31,"endTime":560.67,"body":"different potential outcomes."},{"startTime":560.67,"endTime":563.73,"body":"We'd be crying by the time we did that."},{"startTime":563.73,"endTime":566.31,"body":"So let's instead say how could we lose?"},{"startTime":566.31,"endTime":567.99,"body":"And the only way we can lose"},{"startTime":567.99,"endTime":572.99,"body":"is by failing 24 consecutive"},{"startTime":567.99,"endTime":572.99,"body":"times to throw a double six."},{"startTime":572.997,"endTime":576.6,"body":"Now, the chance of throwing a"},{"startTime":572.997,"endTime":576.6,"body":"double six in any one attempt,"},{"startTime":576.6,"endTime":578.637,"body":"as we've seen, one in 36."},{"startTime":578.637,"endTime":583.637,"body":"So the chance of failing to"},{"startTime":578.637,"endTime":583.637,"body":"do that will be 35/36, okay?"},{"startTime":584.55,"endTime":587.7,"body":"So if we're going to fail"},{"startTime":584.55,"endTime":587.7,"body":"24 consecutive times,"},{"startTime":587.7,"endTime":591.39,"body":"then the chance of that will be 35/36"},{"startTime":591.39,"endTime":593.64,"body":"raised to the power of 24."},{"startTime":593.64,"endTime":595.92,"body":"And then that's the"},{"startTime":593.64,"endTime":595.92,"body":"probability of failure,"},{"startTime":595.92,"endTime":599.01,"body":"so the probability of"},{"startTime":595.92,"endTime":599.01,"body":"success is one minus that."},{"startTime":599.01,"endTime":601.32,"body":"Brilliant, so we just put"},{"startTime":599.01,"endTime":601.32,"body":"that in our calculator"},{"startTime":601.32,"endTime":602.46,"body":"and work it out."},{"startTime":602.46,"endTime":605.94,"body":"Only Pascal and Fermat do"},{"startTime":602.46,"endTime":605.94,"body":"not have our calculator,"},{"startTime":605.94,"endTime":607.943,"body":"or any calculator, right? So"},{"startTime":605.94,"endTime":607.943,"body":"what are they going to do?"},{"startTime":607.943,"endTime":610.44,"body":"Well, how can they work, this would be,"},{"startTime":610.44,"endTime":612.24,"body":"I don't want to be doing that by hand."},{"startTime":612.24,"endTime":614.07,"body":"Nobody wants to do that by hand."},{"startTime":614.07,"endTime":617.97,"body":"Luckily, luckily, there"},{"startTime":614.07,"endTime":617.97,"body":"is a mathematical result"},{"startTime":617.97,"endTime":620.67,"body":"that can help us do precisely"},{"startTime":617.97,"endTime":620.67,"body":"this kind of calculation."},{"startTime":620.67,"endTime":622.184,"body":"It's really, really useful."},{"startTime":622.184,"endTime":624.69,"body":"And it's called the binomial theorem."},{"startTime":624.69,"endTime":626.897,"body":"And if you've heard or"},{"startTime":624.69,"endTime":626.897,"body":"used a binomial theorem,"},{"startTime":626.897,"endTime":628.29,"body":"you'll be thinking, hang on a minute,"},{"startTime":628.29,"endTime":630.21,"body":"that's got something to do"},{"startTime":628.29,"endTime":630.21,"body":"with multiplying out brackets,"},{"startTime":630.21,"endTime":631.62,"body":"and indeed it does."},{"startTime":631.62,"endTime":636.36,"body":"It tells us how to work out"},{"startTime":631.62,"endTime":636.36,"body":"expressions like A plus B,"},{"startTime":636.36,"endTime":637.65,"body":"all to the power N, right?"},{"startTime":637.65,"endTime":639.78,"body":"So bi for two things"},{"startTime":639.78,"endTime":641.58,"body":"that you've got two terms in your bracket."},{"startTime":641.58,"endTime":645.69,"body":"So you want to multiply out"},{"startTime":641.58,"endTime":645.69,"body":"a bracket by itself N times,"},{"startTime":645.69,"endTime":647.28,"body":"a great big algebraic calculation."},{"startTime":647.28,"endTime":650.1,"body":"And the theorem tells you how to do that,"},{"startTime":650.1,"endTime":652.77,"body":"what the coefficients"},{"startTime":650.1,"endTime":652.77,"body":"are of all those terms."},{"startTime":652.77,"endTime":656.13,"body":"Now, a special example of"},{"startTime":652.77,"endTime":656.13,"body":"this, which is relevant to us,"},{"startTime":656.13,"endTime":657.42,"body":"is something like this."},{"startTime":657.42,"endTime":661.487,"body":"If you wanted to work out one"},{"startTime":657.42,"endTime":661.487,"body":"minus X to the power of 24,"},{"startTime":661.487,"endTime":664.98,"body":"then the theorem will tell"},{"startTime":661.487,"endTime":664.98,"body":"you it's an expression"},{"startTime":664.98,"endTime":668.94,"body":"that looks a bit like that,"},{"startTime":664.98,"endTime":668.94,"body":"some X's, X squared, X cubed,"},{"startTime":668.94,"endTime":670.65,"body":"and so on, higher powers of X."},{"startTime":670.65,"endTime":673.8,"body":"And those coefficients that"},{"startTime":670.65,"endTime":673.8,"body":"I haven't explicitly stated"},{"startTime":673.8,"endTime":675.18,"body":"what I've called U, V and W,"},{"startTime":675.18,"endTime":677.607,"body":"the theorem tells you"},{"startTime":675.18,"endTime":677.607,"body":"exactly how to find those."},{"startTime":677.607,"endTime":680.25,"body":"And we call them binomial coefficients."},{"startTime":680.25,"endTime":683.04,"body":"Now, the useful thing for us here"},{"startTime":683.04,"endTime":688.04,"body":"is that if your X in that"},{"startTime":683.04,"endTime":688.04,"body":"expression, if X is a small number"},{"startTime":688.063,"endTime":692.22,"body":"and X is already small,"},{"startTime":688.063,"endTime":692.22,"body":"X squared is very small,"},{"startTime":692.22,"endTime":697.22,"body":"X cubed is tiny, and every"},{"startTime":692.22,"endTime":697.22,"body":"higher power of X is increasing,"},{"startTime":697.41,"endTime":699.48,"body":"well, it is very, very, very small."},{"startTime":699.48,"endTime":701.25,"body":"And after awhile it's kind of negligible."},{"startTime":701.25,"endTime":703.579,"body":"If you just want an approximate answer,"},{"startTime":703.579,"endTime":706.35,"body":"you can ignore those higher terms in X"},{"startTime":706.35,"endTime":708.75,"body":"'cause they're very, very small."},{"startTime":708.75,"endTime":710.82,"body":"So what's our X going to be?"},{"startTime":710.82,"endTime":715.44,"body":"Well, if we want to work out"},{"startTime":710.82,"endTime":715.44,"body":"35/36 to the power of 24,"},{"startTime":715.44,"endTime":718.44,"body":"that's one minus one over 36."},{"startTime":718.44,"endTime":721.8,"body":"So our X is one over 36, which is small."},{"startTime":721.8,"endTime":723.21,"body":"And so we can just calculate"},{"startTime":723.21,"endTime":725.4,"body":"the first few terms of"},{"startTime":723.21,"endTime":725.4,"body":"this long expression."},{"startTime":725.4,"endTime":727.35,"body":"We don't care about the higher powers."},{"startTime":727.35,"endTime":729.0,"body":"We just have two or three terms,"},{"startTime":729.0,"endTime":731.37,"body":"and it will give us a"},{"startTime":729.0,"endTime":731.37,"body":"good enough approximation."},{"startTime":731.37,"endTime":735.06,"body":"And the approximation"},{"startTime":731.37,"endTime":735.06,"body":"we get is about 0.51."},{"startTime":735.06,"endTime":740.06,"body":"So that's a 51% chance of"},{"startTime":735.06,"endTime":740.06,"body":"failing to throw a double six."},{"startTime":740.94,"endTime":742.86,"body":"The odds are kind of switched over."},{"startTime":742.86,"endTime":746.88,"body":"Gombaud is more likely, so"},{"startTime":742.86,"endTime":746.88,"body":"his chance of succeeding"},{"startTime":746.88,"endTime":749.67,"body":"to throw a double six in 24"},{"startTime":746.88,"endTime":749.67,"body":"attempts is less than half."},{"startTime":749.67,"endTime":751.44,"body":"It's a 49% chance."},{"startTime":751.44,"endTime":753.66,"body":"So actually we should"},{"startTime":751.44,"endTime":753.66,"body":"take that bet, (laughs)"},{"startTime":753.66,"endTime":756.57,"body":"if he asked us to do this,"},{"startTime":753.66,"endTime":756.57,"body":"because he has a 49% chance"},{"startTime":756.57,"endTime":759.918,"body":"only of winning, and we would"},{"startTime":756.57,"endTime":759.918,"body":"have a 51% chance therefore"},{"startTime":759.918,"endTime":762.48,"body":"of getting money from him."},{"startTime":762.48,"endTime":765.21,"body":"So it's wrong that that"},{"startTime":762.48,"endTime":765.21,"body":"little intuition there,"},{"startTime":765.21,"endTime":767.43,"body":"the old gambler's rule, didn't work out,"},{"startTime":767.43,"endTime":768.75,"body":"and we can prove why."},{"startTime":768.75,"endTime":770.73,"body":"But we have to use something"},{"startTime":768.75,"endTime":770.73,"body":"like the binomial theorem"},{"startTime":770.73,"endTime":771.563,"body":"to do so."},{"startTime":772.68,"endTime":775.8,"body":"I will just mention"},{"startTime":772.68,"endTime":775.8,"body":"another of the problems"},{"startTime":775.8,"endTime":778.527,"body":"that Pascal and Fermat"},{"startTime":775.8,"endTime":778.527,"body":"discussed in their letters,"},{"startTime":778.527,"endTime":781.74,"body":"'cause it does have applications today"},{"startTime":781.74,"endTime":784.41,"body":"in people are thinking about"},{"startTime":781.74,"endTime":784.41,"body":"how to bet on sporting events"},{"startTime":784.41,"endTime":785.55,"body":"or something like that."},{"startTime":785.55,"endTime":788.01,"body":"It's known as the problem of points."},{"startTime":788.01,"endTime":792.96,"body":"And the idea is that you've"},{"startTime":788.01,"endTime":792.96,"body":"started playing some game,"},{"startTime":792.96,"endTime":796.11,"body":"you're gambling on something,"},{"startTime":792.96,"endTime":796.11,"body":"dice or coins or something,"},{"startTime":796.11,"endTime":799.5,"body":"rain stops the play, so"},{"startTime":796.11,"endTime":799.5,"body":"the Duckworth-Lewis method,"},{"startTime":799.5,"endTime":802.38,"body":"isn't quite that; it's not"},{"startTime":799.5,"endTime":802.38,"body":"going to be as bad as that."},{"startTime":802.38,"endTime":806.55,"body":"Something stops the game, or"},{"startTime":802.38,"endTime":806.55,"body":"you're at a point in the game,"},{"startTime":806.55,"endTime":810.48,"body":"and you want to know actually"},{"startTime":806.55,"endTime":810.48,"body":"how have the odds changed"},{"startTime":810.48,"endTime":812.25,"body":"based on what's happened so far."},{"startTime":812.25,"endTime":816.12,"body":"So you might go into, say,"},{"startTime":812.25,"endTime":816.12,"body":"you and I are tossing a coin,"},{"startTime":816.12,"endTime":817.5,"body":"and it's the first of three."},{"startTime":817.5,"endTime":820.38,"body":"So maybe I've chosen"},{"startTime":817.5,"endTime":820.38,"body":"heads, you've chosen tails,"},{"startTime":820.38,"endTime":823.71,"body":"and we've got, so far, we've"},{"startTime":820.38,"endTime":823.71,"body":"thrown heads, heads, tails."},{"startTime":823.71,"endTime":826.77,"body":"So I just need one more head;"},{"startTime":823.71,"endTime":826.77,"body":"you need two more tails."},{"startTime":826.77,"endTime":830.07,"body":"So I've clearly got an"},{"startTime":826.77,"endTime":830.07,"body":"advantage at this point."},{"startTime":830.07,"endTime":831.973,"body":"I've got a, clearly going to have"},{"startTime":831.973,"endTime":834.87,"body":"a more than a 50/50 chance of winning now,"},{"startTime":834.87,"endTime":838.05,"body":"but exactly what chance, how should we,"},{"startTime":838.05,"endTime":840.24,"body":"if the game is abandoned at this point,"},{"startTime":840.24,"endTime":844.11,"body":"how do we divvy up, how do"},{"startTime":840.24,"endTime":844.11,"body":"we share the kitty or the pot"},{"startTime":844.11,"endTime":846.12,"body":"if we have to stop at this point?"},{"startTime":846.12,"endTime":849.87,"body":"So you could reason it like this."},{"startTime":849.87,"endTime":851.52,"body":"You could say, there's only three ways"},{"startTime":851.52,"endTime":855.24,"body":"this game can end now, by"},{"startTime":851.52,"endTime":855.24,"body":"if the next row is heads,"},{"startTime":855.24,"endTime":858.72,"body":"we're done, I've won,"},{"startTime":855.24,"endTime":858.72,"body":"'cause I've got three heads."},{"startTime":858.72,"endTime":860.82,"body":"Or the next throw could be tails,"},{"startTime":860.82,"endTime":864.24,"body":"but then if after that it's"},{"startTime":860.82,"endTime":864.24,"body":"heads, I've won, again,"},{"startTime":864.24,"endTime":865.83,"body":"'cause I still have my three heads"},{"startTime":865.83,"endTime":867.63,"body":"before you've got your three tails."},{"startTime":867.63,"endTime":871.59,"body":"And so the only way of those"},{"startTime":867.63,"endTime":871.59,"body":"three possible end games"},{"startTime":871.59,"endTime":875.04,"body":"where you would win is if"},{"startTime":871.59,"endTime":875.04,"body":"it's two tails consecutively."},{"startTime":875.04,"endTime":877.743,"body":"So therefore, I should get"},{"startTime":875.04,"endTime":877.743,"body":"two-thirds of the kitty."},{"startTime":878.7,"endTime":882.15,"body":"Well, no; no, again, the issue is"},{"startTime":882.15,"endTime":884.55,"body":"these things are not all equally likely."},{"startTime":884.55,"endTime":886.77,"body":"And what Pascal and"},{"startTime":884.55,"endTime":886.77,"body":"Fermat sort of realized"},{"startTime":886.77,"endTime":890.982,"body":"was that you've got to consider"},{"startTime":890.982,"endTime":894.03,"body":"all the potential things"},{"startTime":890.982,"endTime":894.03,"body":"that could happen,"},{"startTime":894.03,"endTime":898.23,"body":"even if they don't actually"},{"startTime":894.03,"endTime":898.23,"body":"happen during gameplay."},{"startTime":898.23,"endTime":901.74,"body":"And the reason you do that"},{"startTime":898.23,"endTime":901.74,"body":"is so that you can consider"},{"startTime":901.74,"endTime":903.663,"body":"all the events with equal likelihood."},{"startTime":904.533,"endTime":907.529,"body":"So if you toss a coin two times,"},{"startTime":907.529,"endTime":909.63,"body":"there are actually four possible outcomes."},{"startTime":909.63,"endTime":913.8,"body":"Heads then heads, heads then"},{"startTime":909.63,"endTime":913.8,"body":"tails, tails then heads,"},{"startTime":913.8,"endTime":917.46,"body":"or tails then tails, right,"},{"startTime":913.8,"endTime":917.46,"body":"four equally likely outcomes."},{"startTime":917.46,"endTime":919.755,"body":"So if you can, I'll put those in."},{"startTime":919.755,"endTime":921.75,"body":"Now, the first two of those,"},{"startTime":921.75,"endTime":924.78,"body":"you never actually play to"},{"startTime":921.75,"endTime":924.78,"body":"the conclusion of those."},{"startTime":924.78,"endTime":928.29,"body":"Because as soon as you've"},{"startTime":924.78,"endTime":928.29,"body":"thrown that next head there,"},{"startTime":928.29,"endTime":929.73,"body":"the game is over."},{"startTime":929.73,"endTime":931.47,"body":"But you still have to"},{"startTime":929.73,"endTime":931.47,"body":"include those sort of"},{"startTime":931.47,"endTime":935.85,"body":"shadow ghost potential"},{"startTime":931.47,"endTime":935.85,"body":"universes in your calculation,"},{"startTime":935.85,"endTime":940.05,"body":"even though no game could"},{"startTime":935.85,"endTime":940.05,"body":"list that sequence of throws"},{"startTime":940.05,"endTime":942.456,"body":"because you would stop"},{"startTime":940.05,"endTime":942.456,"body":"before you get to the end."},{"startTime":942.456,"endTime":944.34,"body":"When you do that, when you list"},{"startTime":944.34,"endTime":946.35,"body":"even the sort of potential outcomes,"},{"startTime":946.35,"endTime":949.11,"body":"even if they may not happen in gameplay,"},{"startTime":949.11,"endTime":950.73,"body":"it's then very easy to see, okay,"},{"startTime":950.73,"endTime":953.85,"body":"there are four of these"},{"startTime":950.73,"endTime":953.85,"body":"all equally likely."},{"startTime":953.85,"endTime":957.015,"body":"In three of them, I win;"},{"startTime":953.85,"endTime":957.015,"body":"I've rigged this, clearly,"},{"startTime":957.015,"endTime":958.95,"body":"(laughs) in three of them,"},{"startTime":957.015,"endTime":958.95,"body":"I win, so I should get"},{"startTime":958.95,"endTime":960.57,"body":"three-quarters of the kitty,"},{"startTime":960.57,"endTime":962.992,"body":"if we have to stop at this point."},{"startTime":962.992,"endTime":967.8,"body":"Okay, so that's the kind of"},{"startTime":962.992,"endTime":967.8,"body":"deduction that they were making."},{"startTime":967.8,"endTime":971.7,"body":"Now, this kind of reasoning"},{"startTime":971.7,"endTime":973.74,"body":"was really one of the very early attempts"},{"startTime":973.74,"endTime":976.012,"body":"to properly lay down some foundations"},{"startTime":976.012,"endTime":977.82,"body":"of the theory of probability,"},{"startTime":977.82,"endTime":980.58,"body":"and it went from strength to strength."},{"startTime":980.58,"endTime":984.39,"body":"But I want to give you"},{"startTime":980.58,"endTime":984.39,"body":"a couple of examples"},{"startTime":984.39,"endTime":986.43,"body":"of a general principle, really,"},{"startTime":986.43,"endTime":989.16,"body":"that human beings as a species,"},{"startTime":989.16,"endTime":994.16,"body":"we are poor at probabilistic intuition."},{"startTime":994.41,"endTime":997.2,"body":"We don't seem to have very good intuition"},{"startTime":997.2,"endTime":1002.06,"body":"when it comes to risk and"},{"startTime":997.2,"endTime":1002.06,"body":"chance, and these kind of things."},{"startTime":1002.06,"endTime":1004.718,"body":"And I just wanted, this is"},{"startTime":1002.06,"endTime":1004.718,"body":"not, I'm not saying this"},{"startTime":1004.718,"endTime":1008.33,"body":"to crush our self-esteem,"},{"startTime":1004.718,"endTime":1008.33,"body":"but knowledge is power."},{"startTime":1008.33,"endTime":1010.49,"body":"And so if we know this about ourselves,"},{"startTime":1010.49,"endTime":1013.64,"body":"we know we're not great"},{"startTime":1010.49,"endTime":1013.64,"body":"interpreting statistics,"},{"startTime":1013.64,"endTime":1015.74,"body":"then it just means we can be"},{"startTime":1013.64,"endTime":1015.74,"body":"a bit more vigilant, right?"},{"startTime":1015.74,"endTime":1017.33,"body":"So to make us feel better about"},{"startTime":1017.33,"endTime":1019.73,"body":"our own imperfection as humans,"},{"startTime":1019.73,"endTime":1022.4,"body":"I want to give you just two examples"},{"startTime":1022.4,"endTime":1025.13,"body":"of really pretty good mathematicians"},{"startTime":1025.13,"endTime":1029.12,"body":"who made what we might call"},{"startTime":1025.13,"endTime":1029.12,"body":"basic errors of probability."},{"startTime":1029.12,"endTime":1031.85,"body":"And this is not me saying that"},{"startTime":1029.12,"endTime":1031.85,"body":"they're bad mathematicians."},{"startTime":1031.85,"endTime":1033.26,"body":"They are great mathematicians,"},{"startTime":1033.26,"endTime":1034.55,"body":"and so it's even more surprising"},{"startTime":1034.55,"endTime":1036.38,"body":"to say that they would make some errors."},{"startTime":1036.38,"endTime":1039.53,"body":"So here are two questions,"},{"startTime":1036.38,"endTime":1039.53,"body":"right, that I'm going to give you"},{"startTime":1039.53,"endTime":1042.29,"body":"some surprisingly incorrect answers to."},{"startTime":1042.29,"endTime":1044.96,"body":"The first one; if a coin is tossed twice,"},{"startTime":1044.96,"endTime":1046.82,"body":"we've just talked about this actually,"},{"startTime":1046.82,"endTime":1049.703,"body":"what's the probability of"},{"startTime":1046.82,"endTime":1049.703,"body":"getting at least one head?"},{"startTime":1050.63,"endTime":1054.35,"body":"And the second question,"},{"startTime":1050.63,"endTime":1054.35,"body":"throwing two dice,"},{"startTime":1054.35,"endTime":1058.526,"body":"which total is more"},{"startTime":1054.35,"endTime":1058.526,"body":"likely to get, 11 or 12?"},{"startTime":1058.526,"endTime":1062.84,"body":"Okay, so first one on the"},{"startTime":1058.526,"endTime":1062.84,"body":"left, let us hear the answer"},{"startTime":1062.84,"endTime":1066.44,"body":"of extremely good mathematician,"},{"startTime":1062.84,"endTime":1066.44,"body":"Jean Le Rond d'Alembert,"},{"startTime":1066.44,"endTime":1068.75,"body":"who wrote an Encyclopedia"},{"startTime":1066.44,"endTime":1068.75,"body":"article about it,"},{"startTime":1068.75,"endTime":1071.84,"body":"this is after the time"},{"startTime":1068.75,"endTime":1071.84,"body":"of Fermat and Pascal."},{"startTime":1071.84,"endTime":1074.247,"body":"Here he is; okay, he"},{"startTime":1071.84,"endTime":1074.247,"body":"says, \"The possibility is"},{"startTime":1074.247,"endTime":1075.897,"body":"\"if you're trying to"},{"startTime":1074.247,"endTime":1075.897,"body":"get at least one head,"},{"startTime":1075.897,"endTime":1077.817,"body":"\"well, you could throw a head straightaway"},{"startTime":1077.817,"endTime":1080.817,"body":"\"or you carry on trying,"},{"startTime":1077.817,"endTime":1080.817,"body":"and it could be tails,"},{"startTime":1080.817,"endTime":1083.24,"body":"\"but then heads, or it"},{"startTime":1080.817,"endTime":1083.24,"body":"could be two tails.\""},{"startTime":1083.24,"endTime":1084.767,"body":"So three possibilities."},{"startTime":1084.767,"endTime":1086.54,"body":"Two-thirds of the time you get a a head,"},{"startTime":1086.54,"endTime":1088.4,"body":"so the probability is two-thirds, right?"},{"startTime":1088.4,"endTime":1092.51,"body":"It's that same incorrect"},{"startTime":1088.4,"endTime":1092.51,"body":"thing that we talked about"},{"startTime":1092.51,"endTime":1095.999,"body":"a moment ago, the real"},{"startTime":1092.51,"endTime":1095.999,"body":"answer, no, is three-quarters."},{"startTime":1095.999,"endTime":1099.56,"body":"And it's again because he"},{"startTime":1095.999,"endTime":1099.56,"body":"didn't consider outcomes"},{"startTime":1099.56,"endTime":1102.86,"body":"that you would never"},{"startTime":1099.56,"endTime":1102.86,"body":"play to their completion,"},{"startTime":1102.86,"endTime":1104.51,"body":"because you already stop"},{"startTime":1102.86,"endTime":1104.51,"body":"when you get a head."},{"startTime":1104.51,"endTime":1107.12,"body":"So now this guy is a really"},{"startTime":1104.51,"endTime":1107.12,"body":"amazing mathematician."},{"startTime":1107.12,"endTime":1110.18,"body":"If you ever have done or"},{"startTime":1107.12,"endTime":1110.18,"body":"will do a math degree,"},{"startTime":1110.18,"endTime":1112.082,"body":"you will hear about"},{"startTime":1110.18,"endTime":1112.082,"body":"some of his mathematics."},{"startTime":1112.082,"endTime":1115.46,"body":"But he made this silly"},{"startTime":1112.082,"endTime":1115.46,"body":"mistake, you know, even he."},{"startTime":1115.46,"endTime":1120.46,"body":"And here's another amazingly,"},{"startTime":1115.46,"endTime":1120.46,"body":"right, okay, Leibniz."},{"startTime":1120.86,"endTime":1122.39,"body":"What a guy, right?"},{"startTime":1122.39,"endTime":1124.34,"body":"Invented calculus with Newton."},{"startTime":1124.34,"endTime":1125.78,"body":"He has a biscuit named after him."},{"startTime":1125.78,"endTime":1127.13,"body":"That is how"},{"startTime":1125.78,"endTime":1127.13,"body":"(students laugh)"},{"startTime":1127.13,"endTime":1128.96,"body":"amazing he is as a mathematician."},{"startTime":1128.96,"endTime":1130.61,"body":"We can only dream of such fame."},{"startTime":1130.61,"endTime":1135.61,"body":"So Choco Leibniz here"},{"startTime":1130.61,"endTime":1135.61,"body":"says, \"It's equally likely"},{"startTime":1135.704,"endTime":1140.247,"body":"\"to throw 12 points as 11,"},{"startTime":1135.704,"endTime":1140.247,"body":"because,\" what does he say?"},{"startTime":1140.247,"endTime":1143.12,"body":"\"One or other can be"},{"startTime":1140.247,"endTime":1143.12,"body":"done in only one manner.\""},{"startTime":1143.12,"endTime":1146.3,"body":"So what he means here is,"},{"startTime":1143.12,"endTime":1146.3,"body":"okay, to throw a total of 12"},{"startTime":1146.3,"endTime":1149.81,"body":"with two dice, they"},{"startTime":1146.3,"endTime":1149.81,"body":"both have to be six, eh?"},{"startTime":1149.81,"endTime":1151.873,"body":"To throw an 11, you've got"},{"startTime":1149.81,"endTime":1151.873,"body":"to have one five and one six."},{"startTime":1151.873,"endTime":1154.04,"body":"There's only one way to do it."},{"startTime":1154.04,"endTime":1158.0,"body":"But that's not, (laughs)"},{"startTime":1154.04,"endTime":1158.0,"body":"that's not quite right."},{"startTime":1158.0,"endTime":1161.03,"body":"Temporarily, you have to care"},{"startTime":1161.03,"endTime":1162.32,"body":"that you have two different dice,"},{"startTime":1162.32,"endTime":1165.118,"body":"and you have to care which one is which."},{"startTime":1165.118,"endTime":1168.527,"body":"Because you could have, the"},{"startTime":1165.118,"endTime":1168.527,"body":"one on the left could be five"},{"startTime":1168.527,"endTime":1170.81,"body":"and the one on the right could be six,"},{"startTime":1170.81,"endTime":1171.977,"body":"or the one on the left could be six"},{"startTime":1171.977,"endTime":1175.277,"body":"and the one on the right"},{"startTime":1171.977,"endTime":1175.277,"body":"could be five, right?"},{"startTime":1175.277,"endTime":1176.42,"body":"Oh, I said that right?"},{"startTime":1176.42,"endTime":1178.82,"body":"Anyway, so you could have"},{"startTime":1176.42,"endTime":1178.82,"body":"five and six or six and five,"},{"startTime":1178.82,"endTime":1180.95,"body":"and those are actually different"},{"startTime":1180.95,"endTime":1182.75,"body":"temporarily from your"},{"startTime":1180.95,"endTime":1182.75,"body":"point of view, right?"},{"startTime":1182.75,"endTime":1184.73,"body":"So there are actually"},{"startTime":1182.75,"endTime":1184.73,"body":"two ways to throw an 11."},{"startTime":1184.73,"endTime":1187.64,"body":"It's twice as likely to get a total of 11"},{"startTime":1187.64,"endTime":1188.9,"body":"as it is to get 12."},{"startTime":1188.9,"endTime":1191.18,"body":"And you know, even the great Leibniz"},{"startTime":1191.18,"endTime":1192.44,"body":"could make a little mistake like that."},{"startTime":1192.44,"endTime":1194.84,"body":"So just you know, watch out."},{"startTime":1194.84,"endTime":1197.9,"body":"The biggest, the biggest"},{"startTime":1194.84,"endTime":1197.9,"body":"mistake that we all,"},{"startTime":1197.9,"endTime":1200.96,"body":"it's very tempting to do,"},{"startTime":1197.9,"endTime":1200.96,"body":"it's a beguiling error,"},{"startTime":1200.96,"endTime":1202.947,"body":"is to say something like, you know,"},{"startTime":1202.947,"endTime":1206.337,"body":"\"I've thrown four heads in a row,"},{"startTime":1206.337,"endTime":1209.18,"body":"\"so I'm overdue to get a tails.\""},{"startTime":1209.18,"endTime":1211.68,"body":"And you sort of feel like"},{"startTime":1209.18,"endTime":1211.68,"body":"the universe owes you"},{"startTime":1212.663,"endTime":1213.496,"body":"(laughs) the other outcome."},{"startTime":1213.496,"endTime":1214.329,"body":"Because you know that"},{"startTime":1214.329,"endTime":1216.08,"body":"they're equally likely in the long run,"},{"startTime":1216.08,"endTime":1218.99,"body":"but, of course, coins don't have a memory,"},{"startTime":1218.99,"endTime":1220.55,"body":"roulette tables do not have a memory."},{"startTime":1220.55,"endTime":1222.83,"body":"This is the gambler's fallacy of saying,"},{"startTime":1222.83,"endTime":1224.63,"body":"we know what the odds are in the long run,"},{"startTime":1224.63,"endTime":1226.097,"body":"and therefore, you start to think,"},{"startTime":1226.097,"endTime":1228.02,"body":"oh, I'm definitely going"},{"startTime":1226.097,"endTime":1228.02,"body":"to get it this time."},{"startTime":1228.02,"endTime":1229.64,"body":"This time, it's got to happen."},{"startTime":1229.64,"endTime":1232.76,"body":"So on a roulette table, you"},{"startTime":1229.64,"endTime":1232.76,"body":"might bet on red and black,"},{"startTime":1232.76,"endTime":1234.74,"body":"those are equally likely to occur."},{"startTime":1234.74,"endTime":1236.63,"body":"It just feels a slightly less than 50%,"},{"startTime":1236.63,"endTime":1237.65,"body":"because there's actually"},{"startTime":1237.65,"endTime":1240.8,"body":"a sneaky green one in there as well."},{"startTime":1240.8,"endTime":1242.51,"body":"That's how the casinos make their money."},{"startTime":1242.51,"endTime":1247.51,"body":"This is a beguiling"},{"startTime":1242.51,"endTime":1247.51,"body":"fallacy; it isn't true."},{"startTime":1248.504,"endTime":1251.45,"body":"The roulette table, the dice, the coins,"},{"startTime":1251.45,"endTime":1253.13,"body":"they don't remember what's happened."},{"startTime":1253.13,"endTime":1256.76,"body":"They don't have a sense"},{"startTime":1253.13,"endTime":1256.76,"body":"od moral justice. (laughs)"},{"startTime":1256.76,"endTime":1258.35,"body":"It's randomness."},{"startTime":1258.35,"endTime":1260.84,"body":"And the point is, any,"},{"startTime":1260.84,"endTime":1261.95,"body":"if you're talking about tossing coins,"},{"startTime":1261.95,"endTime":1266.95,"body":"any sequence of heads and"},{"startTime":1261.95,"endTime":1266.95,"body":"tails that you specify exactly"},{"startTime":1267.26,"endTime":1270.05,"body":"is equally likely as any other."},{"startTime":1270.05,"endTime":1273.26,"body":"If you have a sequence of six coin tosses,"},{"startTime":1273.26,"endTime":1277.01,"body":"there's a 50/50 chance at"},{"startTime":1273.26,"endTime":1277.01,"body":"each stage of heads or tails."},{"startTime":1277.01,"endTime":1279.47,"body":"So you multiply those halves"},{"startTime":1277.01,"endTime":1279.47,"body":"together, six of them,"},{"startTime":1279.47,"endTime":1281.36,"body":"you get a one in 64 chance"},{"startTime":1281.36,"endTime":1284.21,"body":"of any particular specified sequence."},{"startTime":1284.21,"endTime":1288.86,"body":"So it's a one in 64 chance of"},{"startTime":1284.21,"endTime":1288.86,"body":"getting six heads in a row,"},{"startTime":1288.86,"endTime":1290.99,"body":"but there's also a one in 64 chance"},{"startTime":1290.99,"endTime":1293.3,"body":"of getting five heads then a tail."},{"startTime":1293.3,"endTime":1296.0,"body":"And you don't know"},{"startTime":1293.3,"endTime":1296.0,"body":"which universe you're in"},{"startTime":1296.0,"endTime":1297.32,"body":"when you've had five heads."},{"startTime":1297.32,"endTime":1298.43,"body":"You don't know if you're in the universe"},{"startTime":1298.43,"endTime":1301.58,"body":"of five heads then another"},{"startTime":1298.43,"endTime":1301.58,"body":"head, or five heads and a tail."},{"startTime":1301.58,"endTime":1303.98,"body":"At that point, they are equally likely."},{"startTime":1303.98,"endTime":1306.89,"body":"And so you cannot do this"},{"startTime":1303.98,"endTime":1306.89,"body":"kind of, \"Oh, I'm overdue\""},{"startTime":1306.89,"endTime":1309.05,"body":"a head or a black or a white."},{"startTime":1309.05,"endTime":1313.55,"body":"However, here is my"},{"startTime":1309.05,"endTime":1313.55,"body":"top guaranteed strategy"},{"startTime":1313.55,"endTime":1315.56,"body":"to always win at roulette, right?"},{"startTime":1315.56,"endTime":1316.67,"body":"You ready?"},{"startTime":1316.67,"endTime":1318.65,"body":"This is amazing."},{"startTime":1318.65,"endTime":1320.956,"body":"Free to you, 'cause I love you so much."},{"startTime":1320.956,"endTime":1323.48,"body":"Right, what you do is you put,"},{"startTime":1323.48,"endTime":1325.52,"body":"your initial bet is, say, $1, right?"},{"startTime":1325.52,"endTime":1328.701,"body":"I don't know why I decided,"},{"startTime":1325.52,"endTime":1328.701,"body":"dollars felt more gambly."},{"startTime":1328.701,"endTime":1331.4,"body":"(laughs) Anyway, the initial bet is $1."},{"startTime":1331.4,"endTime":1332.48,"body":"And so when you're doing this,"},{"startTime":1332.48,"endTime":1334.372,"body":"when you're betting on"},{"startTime":1332.48,"endTime":1334.372,"body":"red or black in roulette,"},{"startTime":1334.372,"endTime":1337.79,"body":"if you bet a dollar and you're correct,"},{"startTime":1337.79,"endTime":1339.35,"body":"then you would get $2 back,"},{"startTime":1339.35,"endTime":1341.263,"body":"so you sort of double your money."},{"startTime":1341.263,"endTime":1343.43,"body":"You get your bet back and"},{"startTime":1341.263,"endTime":1343.43,"body":"the same amount, right?"},{"startTime":1343.43,"endTime":1347.15,"body":"So my strategy is, you keep going."},{"startTime":1347.15,"endTime":1351.17,"body":"Every time you lose, you double"},{"startTime":1347.15,"endTime":1351.17,"body":"your stakes and bet again."},{"startTime":1351.17,"endTime":1352.88,"body":"So let's do a little example."},{"startTime":1352.88,"endTime":1355.97,"body":"Suppose you're betting"},{"startTime":1352.88,"endTime":1355.97,"body":"that black will come up,"},{"startTime":1355.97,"endTime":1360.95,"body":"and it doesn't come up"},{"startTime":1355.97,"endTime":1360.95,"body":"until the seventh go."},{"startTime":1360.95,"endTime":1362.57,"body":"So at that point, let's see,"},{"startTime":1362.57,"endTime":1364.22,"body":"what are we betting by the seventh go?"},{"startTime":1364.22,"endTime":1368.578,"body":"So we've got $1, two, then"},{"startTime":1364.22,"endTime":1368.578,"body":"four, then eight, then 16,"},{"startTime":1368.578,"endTime":1370.46,"body":"then 32, 64."},{"startTime":1370.46,"endTime":1374.51,"body":"So we bet $64 then on the"},{"startTime":1370.46,"endTime":1374.51,"body":"time we finally do win,"},{"startTime":1374.51,"endTime":1378.29,"body":"so our winnings will be $128, double that."},{"startTime":1378.29,"endTime":1379.97,"body":"How much have we actually spent?"},{"startTime":1379.97,"endTime":1384.92,"body":"Well, it's all that one, two,"},{"startTime":1379.97,"endTime":1384.92,"body":"four, eight, 16, 32 and 64,"},{"startTime":1384.92,"endTime":1386.317,"body":"that's our outlay."},{"startTime":1386.317,"endTime":1391.317,"body":"And it's a fact about"},{"startTime":1386.317,"endTime":1391.317,"body":"these kind of progressions,"},{"startTime":1391.79,"endTime":1394.76,"body":"if you add up all the powers"},{"startTime":1391.79,"endTime":1394.76,"body":"of two to a certain number,"},{"startTime":1394.76,"endTime":1398.48,"body":"to a certain point, then"},{"startTime":1394.76,"endTime":1398.48,"body":"the sum of all of those"},{"startTime":1398.48,"endTime":1401.273,"body":"is the next power, take away one."},{"startTime":1402.68,"endTime":1405.17,"body":"So the sum of all of those up to 64"},{"startTime":1405.17,"endTime":1409.52,"body":"will be the next power 128 minus one, 127."},{"startTime":1409.52,"endTime":1414.47,"body":"So we actually, our profit,"},{"startTime":1409.52,"endTime":1414.47,"body":"we have made a profit,"},{"startTime":1414.47,"endTime":1417.44,"body":"but it's always just $1."},{"startTime":1417.44,"endTime":1419.93,"body":"It's the same as our initial bet."},{"startTime":1419.93,"endTime":1422.9,"body":"Now, you can address this and say, fine,"},{"startTime":1422.9,"endTime":1425.27,"body":"my initial bet will be a"},{"startTime":1422.9,"endTime":1425.27,"body":"million dollars, right?"},{"startTime":1425.27,"endTime":1427.37,"body":"So then I'm going to definitely"},{"startTime":1425.27,"endTime":1427.37,"body":"make a million dollars profit."},{"startTime":1427.37,"endTime":1430.04,"body":"That's great, except, well,"},{"startTime":1430.04,"endTime":1431.06,"body":"you have to have a million dollars,"},{"startTime":1431.06,"endTime":1432.86,"body":"but you have to have more"},{"startTime":1431.06,"endTime":1432.86,"body":"than a million dollars, right?"},{"startTime":1432.86,"endTime":1435.08,"body":"Because you could go, and"},{"startTime":1432.86,"endTime":1435.08,"body":"this is not by any means"},{"startTime":1435.08,"endTime":1436.82,"body":"unusual enough to not worry about,"},{"startTime":1436.82,"endTime":1439.79,"body":"that you would have to go six"},{"startTime":1436.82,"endTime":1439.79,"body":"or seven steps or even more."},{"startTime":1439.79,"endTime":1441.14,"body":"And by the time you get to that point,"},{"startTime":1441.14,"endTime":1442.86,"body":"how much money have we laid out"},{"startTime":1443.883,"endTime":1445.97,"body":"if we put a million dollars initially?"},{"startTime":1445.97,"endTime":1448.88,"body":"Well, in $127 million investment"},{"startTime":1448.88,"endTime":1450.8,"body":"to only get a million dollars back."},{"startTime":1450.8,"endTime":1453.38,"body":"And at any point, you"},{"startTime":1450.8,"endTime":1453.38,"body":"could run out of money,"},{"startTime":1453.38,"endTime":1454.4,"body":"the casino could say"},{"startTime":1454.4,"endTime":1456.41,"body":"you've reached the"},{"startTime":1454.4,"endTime":1456.41,"body":"maximum bet you can place,"},{"startTime":1456.41,"endTime":1457.243,"body":"or they can just say,"},{"startTime":1457.243,"endTime":1460.018,"body":"\"We don't like the look of you, go away.\""},{"startTime":1460.018,"endTime":1462.98,"body":"So this is perhaps quite a risky strategy,"},{"startTime":1462.98,"endTime":1464.96,"body":"but it does work in theory."},{"startTime":1464.96,"endTime":1466.85,"body":"And I'm sure they wouldn't"},{"startTime":1464.96,"endTime":1466.85,"body":"mind if you do it with $1."},{"startTime":1466.85,"endTime":1470.72,"body":"They may not be too happy if"},{"startTime":1466.85,"endTime":1470.72,"body":"you're betting 64 million."},{"startTime":1470.72,"endTime":1472.55,"body":"If you've got $64 million,"},{"startTime":1472.55,"endTime":1475.28,"body":"you don't need to be wasting your time."},{"startTime":1475.28,"endTime":1478.34,"body":"Just have a lovely time;"},{"startTime":1475.28,"endTime":1478.34,"body":"don't worry about gambling."},{"startTime":1478.34,"endTime":1482.54,"body":"So you do sometimes hear of mathematicians"},{"startTime":1483.92,"endTime":1487.94,"body":"or mathematically-minded people"},{"startTime":1483.92,"endTime":1487.94,"body":"winning money at casinos."},{"startTime":1487.94,"endTime":1492.94,"body":"It's almost always by"},{"startTime":1487.94,"endTime":1492.94,"body":"playing this game, blackjack."},{"startTime":1493.52,"endTime":1496.01,"body":"Now, in blackjack, what happens is"},{"startTime":1496.01,"endTime":1497.72,"body":"you're playing against the dealer,"},{"startTime":1497.72,"endTime":1500.54,"body":"you each get two cards,"},{"startTime":1497.72,"endTime":1500.54,"body":"and then you can decide"},{"startTime":1500.54,"endTime":1502.73,"body":"whether to take more cards at any point"},{"startTime":1502.73,"endTime":1504.29,"body":"or to stick with what you've got."},{"startTime":1504.29,"endTime":1507.05,"body":"And the goal is, you want"},{"startTime":1504.29,"endTime":1507.05,"body":"to get to a total of,"},{"startTime":1507.05,"endTime":1511.517,"body":"as near as you can to 21,"},{"startTime":1507.05,"endTime":1511.517,"body":"but not go over, not go bust."},{"startTime":1511.517,"endTime":1515.69,"body":"So at any point, so here you"},{"startTime":1511.517,"endTime":1515.69,"body":"have a total of 16, right?"},{"startTime":1515.69,"endTime":1519.44,"body":"So you have to think, okay,"},{"startTime":1515.69,"endTime":1519.44,"body":"do I want to take another card"},{"startTime":1519.44,"endTime":1520.91,"body":"and try and get closer to 21?"},{"startTime":1520.91,"endTime":1524.57,"body":"But then the risk is, maybe"},{"startTime":1520.91,"endTime":1524.57,"body":"I take this and it's a nine,"},{"startTime":1524.57,"endTime":1525.89,"body":"and then I've gone bust."},{"startTime":1525.89,"endTime":1528.44,"body":"And the dealer is doing the same thing."},{"startTime":1528.44,"endTime":1530.51,"body":"If they go bust, then you win by default"},{"startTime":1530.51,"endTime":1533.06,"body":"however far away you are."},{"startTime":1533.06,"endTime":1535.64,"body":"But the dealer has to"},{"startTime":1533.06,"endTime":1535.64,"body":"obey the following rule,"},{"startTime":1535.64,"endTime":1537.62,"body":"that if they are on 16 or less,"},{"startTime":1537.62,"endTime":1539.72,"body":"they have to take another card."},{"startTime":1539.72,"endTime":1544.72,"body":"So it is in your interests for"},{"startTime":1539.72,"endTime":1544.72,"body":"the play to reach the point"},{"startTime":1545.9,"endTime":1549.23,"body":"where lots of small cards"},{"startTime":1545.9,"endTime":1549.23,"body":"have already been played."},{"startTime":1549.23,"endTime":1551.18,"body":"And so if that happens,"},{"startTime":1551.18,"endTime":1553.1,"body":"and this is what the"},{"startTime":1551.18,"endTime":1553.1,"body":"card counters are doing,"},{"startTime":1553.1,"endTime":1555.26,"body":"they're monitoring the"},{"startTime":1553.1,"endTime":1555.26,"body":"number of small cards"},{"startTime":1555.26,"endTime":1556.22,"body":"that are still left in play."},{"startTime":1556.22,"endTime":1558.77,"body":"If there are very few, then the chance"},{"startTime":1558.77,"endTime":1560.69,"body":"that the dealer will go bust gets higher."},{"startTime":1560.69,"endTime":1563.33,"body":"So they can sort of gauge"},{"startTime":1560.69,"endTime":1563.33,"body":"the how good the odds are"},{"startTime":1563.33,"endTime":1565.03,"body":"and change their bets accordingly."},{"startTime":1566.24,"endTime":1570.65,"body":"This method, it's, I"},{"startTime":1566.24,"endTime":1570.65,"body":"mean yes, it could work."},{"startTime":1570.65,"endTime":1573.47,"body":"At its best, if you're doing"},{"startTime":1570.65,"endTime":1573.47,"body":"it absolutely perfectly,"},{"startTime":1573.47,"endTime":1578.47,"body":"you get, at best, a 1%"},{"startTime":1573.47,"endTime":1578.47,"body":"advantage over the house casino."},{"startTime":1578.63,"endTime":1580.88,"body":"Any mistake and you risk losing all that."},{"startTime":1580.88,"endTime":1583.67,"body":"Also, you can't be"},{"startTime":1580.88,"endTime":1583.67,"body":"winning too ostentatiously"},{"startTime":1583.67,"endTime":1587.78,"body":"because they're not compelled"},{"startTime":1583.67,"endTime":1587.78,"body":"to have you as a customer."},{"startTime":1587.78,"endTime":1591.05,"body":"So although people have been"},{"startTime":1587.78,"endTime":1591.05,"body":"able to win money at this,"},{"startTime":1591.05,"endTime":1592.313,"body":"it's getting harder."},{"startTime":1592.313,"endTime":1593.69,"body":"One way it's getting harder"},{"startTime":1593.69,"endTime":1595.94,"body":"is that the casinos are"},{"startTime":1593.69,"endTime":1595.94,"body":"adding more decks of cards"},{"startTime":1595.94,"endTime":1597.73,"body":"into the mix. So you"},{"startTime":1595.94,"endTime":1597.73,"body":"know, a long time ago,"},{"startTime":1597.73,"endTime":1599.09,"body":"you might just have one deck of cards,"},{"startTime":1599.09,"endTime":1601.31,"body":"and then you'd play and play"},{"startTime":1599.09,"endTime":1601.31,"body":"until they were all gone,"},{"startTime":1601.31,"endTime":1603.71,"body":"and then you'd start again"},{"startTime":1601.31,"endTime":1603.71,"body":"from the beginning of the deck."},{"startTime":1603.71,"endTime":1605.513,"body":"So it was sort of easier"},{"startTime":1605.513,"endTime":1607.94,"body":"for changes to make more of a difference."},{"startTime":1607.94,"endTime":1610.49,"body":"Now they can use up to eight"},{"startTime":1607.94,"endTime":1610.49,"body":"decks of cards at once,"},{"startTime":1610.49,"endTime":1613.28,"body":"so the variants in that"},{"startTime":1610.49,"endTime":1613.28,"body":"becomes big enough."},{"startTime":1613.28,"endTime":1616.613,"body":"It's harder to make this pay."},{"startTime":1617.739,"endTime":1622.55,"body":"But this precisely is working,"},{"startTime":1617.739,"endTime":1622.55,"body":"because it's not random."},{"startTime":1622.55,"endTime":1624.62,"body":"So what I'm really talking"},{"startTime":1622.55,"endTime":1624.62,"body":"about today is randomness,"},{"startTime":1624.62,"endTime":1626.39,"body":"lotteries, dice and things."},{"startTime":1626.39,"endTime":1629.3,"body":"This is only possible because"},{"startTime":1626.39,"endTime":1629.3,"body":"there isn't randomness,"},{"startTime":1629.3,"endTime":1631.76,"body":"it's not independent of what comes next,"},{"startTime":1631.76,"endTime":1633.89,"body":"because you've been looking at the cards"},{"startTime":1633.89,"endTime":1635.78,"body":"that have already been dealt."},{"startTime":1635.78,"endTime":1638.66,"body":"So I'm not going to talk"},{"startTime":1635.78,"endTime":1638.66,"body":"any more about it here,"},{"startTime":1638.66,"endTime":1642.233,"body":"but that is precisely possible"},{"startTime":1638.66,"endTime":1642.233,"body":"because of non-randomness."},{"startTime":1643.25,"endTime":1645.26,"body":"Okay, but you know, I've"},{"startTime":1643.25,"endTime":1645.26,"body":"promised you lotteries,"},{"startTime":1645.26,"endTime":1646.73,"body":"so let's talk about lotteries."},{"startTime":1646.73,"endTime":1650.6,"body":"The earliest lottery, I think, in England"},{"startTime":1650.6,"endTime":1654.71,"body":"was a state lottery that"},{"startTime":1650.6,"endTime":1654.71,"body":"Elizabeth the First decreed"},{"startTime":1654.71,"endTime":1657.86,"body":"there'd be a lottery, it was"},{"startTime":1654.71,"endTime":1657.86,"body":"to raise money for the Navy."},{"startTime":1657.86,"endTime":1659.63,"body":"Tickets were 10 shillings each."},{"startTime":1659.63,"endTime":1661.73,"body":"That is a lot of money"},{"startTime":1659.63,"endTime":1661.73,"body":"in Elizabethan time."},{"startTime":1661.73,"endTime":1662.78,"body":"So this is not for everyone."},{"startTime":1662.78,"endTime":1666.02,"body":"You have to be pretty"},{"startTime":1662.78,"endTime":1666.02,"body":"well-to-do even to buy a ticket."},{"startTime":1666.02,"endTime":1670.82,"body":"The prize was astronomical, 5,000 pounds,"},{"startTime":1670.82,"endTime":1673.61,"body":"but not all in ready cash."},{"startTime":1673.61,"endTime":1678.61,"body":"You got some tapestry, you got"},{"startTime":1673.61,"endTime":1678.61,"body":"some plates, and some linen,"},{"startTime":1678.95,"endTime":1681.53,"body":"fine linen, oh, well, it"},{"startTime":1678.95,"endTime":1681.53,"body":"better be fine linen cloth,"},{"startTime":1681.53,"endTime":1684.77,"body":"in the mix, so it's kind"},{"startTime":1681.53,"endTime":1684.77,"body":"of an interesting prize."},{"startTime":1684.77,"endTime":1686.93,"body":"Cobbled together"},{"startTime":1684.77,"endTime":1686.93,"body":"everything they could find,"},{"startTime":1686.93,"endTime":1688.19,"body":"ergo, this is what you'll get."},{"startTime":1688.19,"endTime":1689.33,"body":"So this was supposed to raise money."},{"startTime":1689.33,"endTime":1691.13,"body":"I don't think it actually quite succeeded,"},{"startTime":1691.13,"endTime":1695.15,"body":"but that's the first lottery"},{"startTime":1691.13,"endTime":1695.15,"body":"that I'm aware of in England."},{"startTime":1695.15,"endTime":1698.54,"body":"It was more what we'd now"},{"startTime":1695.15,"endTime":1698.54,"body":"call a raffle, actually."},{"startTime":1698.54,"endTime":1701.54,"body":"Modern lotteries, we'll do a"},{"startTime":1698.54,"endTime":1701.54,"body":"sort of little toy example,"},{"startTime":1701.54,"endTime":1704.81,"body":"but modern lotteries all"},{"startTime":1701.54,"endTime":1704.81,"body":"tend to have the form of"},{"startTime":1704.81,"endTime":1709.79,"body":"you choose some numbers from"},{"startTime":1704.81,"endTime":1709.79,"body":"a wider selection of numbers."},{"startTime":1709.79,"endTime":1712.31,"body":"So in the UK, you choose six numbers"},{"startTime":1712.31,"endTime":1713.33,"body":"that are between one and 59."},{"startTime":1713.33,"endTime":1716.333,"body":"It used to be 49, then it went up to 59."},{"startTime":1717.315,"endTime":1720.17,"body":"And then if you match all six numbers,"},{"startTime":1720.17,"endTime":1722.12,"body":"that's how you win the jackpot."},{"startTime":1722.12,"endTime":1724.1,"body":"So we're going to, we'll work up to that,"},{"startTime":1724.1,"endTime":1726.17,"body":"but we'll do a little toy lottery first."},{"startTime":1726.17,"endTime":1729.17,"body":"And now toy lottery, we're"},{"startTime":1726.17,"endTime":1729.17,"body":"choosing from numbers"},{"startTime":1729.17,"endTime":1732.44,"body":"between one and 10, and"},{"startTime":1729.17,"endTime":1732.44,"body":"we choose two numbers."},{"startTime":1732.44,"endTime":1734.6,"body":"And to win the jackpot, we"},{"startTime":1732.44,"endTime":1734.6,"body":"have to match two numbers"},{"startTime":1734.6,"endTime":1737.78,"body":"with what comes out in the exciting draw."},{"startTime":1737.78,"endTime":1741.493,"body":"So what are our odds of"},{"startTime":1737.78,"endTime":1741.493,"body":"winning this toy lottery?"},{"startTime":1741.493,"endTime":1742.326,"body":"So think about this."},{"startTime":1742.326,"endTime":1745.68,"body":"How many ways are there"},{"startTime":1742.326,"endTime":1745.68,"body":"of choosing two numbers"},{"startTime":1745.68,"endTime":1747.613,"body":"from a set of 10? That's"},{"startTime":1745.68,"endTime":1747.613,"body":"what we need to know,"},{"startTime":1747.613,"endTime":1750.5,"body":"because that's the number"},{"startTime":1747.613,"endTime":1750.5,"body":"of possible lottery tickets."},{"startTime":1750.5,"endTime":1753.86,"body":"And there's one winning combination,"},{"startTime":1753.86,"endTime":1755.51,"body":"so out of all the possibilities,"},{"startTime":1755.51,"endTime":1757.66,"body":"we've got a one chance out of all that."},{"startTime":1757.66,"endTime":1759.29,"body":"So how many possible ways are there"},{"startTime":1759.29,"endTime":1761.81,"body":"of picking a set of two numbers out of 10?"},{"startTime":1761.81,"endTime":1764.3,"body":"Well, you imagine the"},{"startTime":1761.81,"endTime":1764.3,"body":"ball there in front of us."},{"startTime":1764.3,"endTime":1765.32,"body":"We pick our first number."},{"startTime":1765.32,"endTime":1766.327,"body":"There's 10 ways to do that,"},{"startTime":1766.327,"endTime":1768.62,"body":"'cause there's 10 numbers in front of us."},{"startTime":1768.62,"endTime":1770.3,"body":"Then we pick our second number."},{"startTime":1770.3,"endTime":1771.619,"body":"Now there's only nine ways to do that,"},{"startTime":1771.619,"endTime":1773.9,"body":"'cause we've already used"},{"startTime":1771.619,"endTime":1773.9,"body":"up one of the numbers."},{"startTime":1773.9,"endTime":1777.95,"body":"So total number of ways to"},{"startTime":1773.9,"endTime":1777.95,"body":"do this is 10 times nine, 90."},{"startTime":1777.95,"endTime":1780.89,"body":"However, however, that forgets"},{"startTime":1780.89,"endTime":1784.16,"body":"that we're actually going to"},{"startTime":1780.89,"endTime":1784.16,"body":"see each combination twice."},{"startTime":1784.16,"endTime":1786.137,"body":"'Cause if, say, you had a one and a five,"},{"startTime":1786.137,"endTime":1787.71,"body":"and that's your winning numbers,"},{"startTime":1787.71,"endTime":1790.61,"body":"well, you could have drawn"},{"startTime":1787.71,"endTime":1790.61,"body":"a one first, then a five,"},{"startTime":1790.61,"endTime":1792.44,"body":"or you could have drawn the"},{"startTime":1790.61,"endTime":1792.44,"body":"five first, then the one."},{"startTime":1792.44,"endTime":1795.53,"body":"Either one of those will"},{"startTime":1792.44,"endTime":1795.53,"body":"result in the same set,"},{"startTime":1795.53,"endTime":1796.94,"body":"one and five."},{"startTime":1796.94,"endTime":1800.93,"body":"So every actual pair is"},{"startTime":1796.94,"endTime":1800.93,"body":"going to appear twice"},{"startTime":1800.93,"endTime":1804.95,"body":"in this selection process"},{"startTime":1800.93,"endTime":1804.95,"body":"of 90 kind of ordered pairs,"},{"startTime":1804.95,"endTime":1806.45,"body":"sequences of two numbers."},{"startTime":1806.45,"endTime":1809.39,"body":"So we've got to divide that 90 by two,"},{"startTime":1809.39,"endTime":1811.52,"body":"and we get, really, there are 45 pairs."},{"startTime":1811.52,"endTime":1815.685,"body":"So it's sort of A, B equals"},{"startTime":1811.52,"endTime":1815.685,"body":"BA in this situation."},{"startTime":1815.685,"endTime":1817.49,"body":"So our chance of winning then,"},{"startTime":1817.49,"endTime":1821.09,"body":"if we want to pick the one"},{"startTime":1817.49,"endTime":1821.09,"body":"that is the jackpot winner,"},{"startTime":1821.09,"endTime":1823.4,"body":"that's a one in 45 chance."},{"startTime":1823.4,"endTime":1827.42,"body":"Okay, step two, let's imagine"},{"startTime":1823.4,"endTime":1827.42,"body":"that's too easy to win."},{"startTime":1827.42,"endTime":1829.1,"body":"Let's now say, okay, for the jackpot,"},{"startTime":1829.1,"endTime":1831.44,"body":"you actually need to match three numbers."},{"startTime":1831.44,"endTime":1834.11,"body":"So now, what are we going to do?"},{"startTime":1834.11,"endTime":1836.6,"body":"How many ways are there of"},{"startTime":1834.11,"endTime":1836.6,"body":"picking a set of three numbers"},{"startTime":1836.6,"endTime":1837.62,"body":"out of 10?"},{"startTime":1837.62,"endTime":1839.18,"body":"Well, you do the same thing."},{"startTime":1839.18,"endTime":1841.565,"body":"You pick the first"},{"startTime":1839.18,"endTime":1841.565,"body":"number, there's 10 ways,"},{"startTime":1841.565,"endTime":1843.89,"body":"then the second number,"},{"startTime":1841.565,"endTime":1843.89,"body":"there's nine ways to do that,"},{"startTime":1843.89,"endTime":1845.75,"body":"nine numbers left, and"},{"startTime":1843.89,"endTime":1845.75,"body":"then the third number,"},{"startTime":1845.75,"endTime":1847.4,"body":"there are now eight numbers left."},{"startTime":1847.4,"endTime":1849.77,"body":"So that's how many"},{"startTime":1847.4,"endTime":1849.77,"body":"choices we've got there."},{"startTime":1849.77,"endTime":1851.36,"body":"So we get 10 times nine times eight,"},{"startTime":1851.36,"endTime":1856.36,"body":"which is 720 of these sequences, right,"},{"startTime":1856.73,"endTime":1860.45,"body":"of numbers A, B, then C, but you"},{"startTime":1856.73,"endTime":1860.45,"body":"again are going to get copies."},{"startTime":1860.45,"endTime":1861.283,"body":"Because it doesn't matter."},{"startTime":1861.283,"endTime":1863.75,"body":"You don't care the order that"},{"startTime":1861.283,"endTime":1863.75,"body":"you drew the numbers out in;"},{"startTime":1863.75,"endTime":1866.03,"body":"you just care what the"},{"startTime":1863.75,"endTime":1866.03,"body":"numbers themselves are."},{"startTime":1866.03,"endTime":1867.11,"body":"So it turns out,"},{"startTime":1867.11,"endTime":1869.54,"body":"and I've written down"},{"startTime":1867.11,"endTime":1869.54,"body":"the possibilities here,"},{"startTime":1869.54,"endTime":1872.84,"body":"there are six ways of rearranging"},{"startTime":1872.84,"endTime":1874.12,"body":"a collection of three things."},{"startTime":1874.12,"endTime":1876.68,"body":"So you can have A, the"},{"startTime":1874.12,"endTime":1876.68,"body":"number A could be first,"},{"startTime":1876.68,"endTime":1880.19,"body":"you could draw that out first,"},{"startTime":1876.68,"endTime":1880.19,"body":"or you could draw B out first"},{"startTime":1880.19,"endTime":1881.48,"body":"or C out first."},{"startTime":1881.48,"endTime":1883.4,"body":"So there's three possibilities"},{"startTime":1883.4,"endTime":1884.81,"body":"for the first number you drew out,"},{"startTime":1884.81,"endTime":1886.34,"body":"because you've got a collection of three"},{"startTime":1886.34,"endTime":1887.373,"body":"that you have to get."},{"startTime":1887.373,"endTime":1890.69,"body":"So there's three ways to get,"},{"startTime":1887.373,"endTime":1890.69,"body":"to have the first number,"},{"startTime":1890.69,"endTime":1893.06,"body":"and then two ways to get the next number,"},{"startTime":1893.06,"endTime":1895.52,"body":"'cause there are two numbers"},{"startTime":1893.06,"endTime":1895.52,"body":"left in your special selection."},{"startTime":1895.52,"endTime":1897.32,"body":"And then the final number,"},{"startTime":1895.52,"endTime":1897.32,"body":"you've got no choice."},{"startTime":1897.32,"endTime":1899.03,"body":"It must be that third one in the set"},{"startTime":1899.03,"endTime":1900.41,"body":"that you're interested in."},{"startTime":1900.41,"endTime":1904.4,"body":"So three times two times one"},{"startTime":1900.41,"endTime":1904.4,"body":"ways to do it, which is six."},{"startTime":1904.4,"endTime":1906.8,"body":"And you may notice, I"},{"startTime":1904.4,"endTime":1906.8,"body":"know I get very excitable,"},{"startTime":1906.8,"endTime":1908.3,"body":"but that exclamation mark"},{"startTime":1908.3,"endTime":1911.54,"body":"is not just me going,"},{"startTime":1908.3,"endTime":1911.54,"body":"\"Hey, three!\" (laughs)"},{"startTime":1911.54,"endTime":1915.71,"body":"That symbol is pronounced"},{"startTime":1911.54,"endTime":1915.71,"body":"in mathematics, \"factorial.\""},{"startTime":1915.71,"endTime":1918.02,"body":"And that three with a"},{"startTime":1915.71,"endTime":1918.02,"body":"little exclamation mark,"},{"startTime":1918.02,"endTime":1920.57,"body":"that means the product of"},{"startTime":1918.02,"endTime":1920.57,"body":"all the numbers up to three."},{"startTime":1920.57,"endTime":1924.74,"body":"And in general, N factorial"},{"startTime":1920.57,"endTime":1924.74,"body":"means multiply all the numbers"},{"startTime":1924.74,"endTime":1927.74,"body":"from one up to N, all the whole numbers."},{"startTime":1927.74,"endTime":1929.45,"body":"So four factorial, for instance,"},{"startTime":1929.45,"endTime":1932.63,"body":"would be one times two"},{"startTime":1929.45,"endTime":1932.63,"body":"times three times four, 24."},{"startTime":1932.63,"endTime":1934.16,"body":"That would be the number of ways"},{"startTime":1934.16,"endTime":1938.0,"body":"of arranging four numbers or objects."},{"startTime":1938.0,"endTime":1940.67,"body":"'Cause you could have any of"},{"startTime":1938.0,"endTime":1940.67,"body":"the four in the first place,"},{"startTime":1940.67,"endTime":1942.89,"body":"then any of the three"},{"startTime":1940.67,"endTime":1942.89,"body":"remaining in the next place,"},{"startTime":1942.89,"endTime":1944.72,"body":"then two and then one."},{"startTime":1944.72,"endTime":1949.473,"body":"Okay, so we have 720"},{"startTime":1944.72,"endTime":1949.473,"body":"sequences of three numbers"},{"startTime":1949.473,"endTime":1954.05,"body":"that we've drawn out,"},{"startTime":1949.473,"endTime":1954.05,"body":"but every collection,"},{"startTime":1954.05,"endTime":1957.53,"body":"every set of three, arises in six ways."},{"startTime":1957.53,"endTime":1958.4,"body":"Those are the six ways."},{"startTime":1958.4,"endTime":1961.25,"body":"So you've got to divide that 720 by six,"},{"startTime":1961.25,"endTime":1963.53,"body":"which I believe is 120."},{"startTime":1963.53,"endTime":1967.4,"body":"So this lottery, you"},{"startTime":1963.53,"endTime":1967.4,"body":"have a one chance in 120"},{"startTime":1967.4,"endTime":1970.34,"body":"of happening upon the correct set."},{"startTime":1970.34,"endTime":1972.17,"body":"All right, so the final little one,"},{"startTime":1972.17,"endTime":1973.43,"body":"now we know how to do this."},{"startTime":1973.43,"endTime":1975.83,"body":"What about choosing four numbers from 10?"},{"startTime":1975.83,"endTime":1980.27,"body":"So a bit of notation, we"},{"startTime":1975.83,"endTime":1980.27,"body":"call that \"10 choose four\""},{"startTime":1980.27,"endTime":1982.46,"body":"that number, and there are"},{"startTime":1980.27,"endTime":1982.46,"body":"different ways of writing it,"},{"startTime":1982.46,"endTime":1985.19,"body":"but my favorite way is you"},{"startTime":1982.46,"endTime":1985.19,"body":"put a 10 and then a four,"},{"startTime":1985.19,"endTime":1987.32,"body":"and then brackets around,"},{"startTime":1985.19,"endTime":1987.32,"body":"so I've done that there."},{"startTime":1987.32,"endTime":1989.12,"body":"So we know how to do this."},{"startTime":1989.12,"endTime":1992.63,"body":"Oh, somehow I've got the"},{"startTime":1989.12,"endTime":1992.63,"body":"last little number missing."},{"startTime":1992.63,"endTime":1994.28,"body":"Looks all right on my screen."},{"startTime":1994.28,"endTime":1996.8,"body":"That should be 10 times"},{"startTime":1994.28,"endTime":1996.8,"body":"nine times eight times seven"},{"startTime":1996.8,"endTime":1998.76,"body":"divided by four times"},{"startTime":1996.8,"endTime":1998.76,"body":"three times two times one,"},{"startTime":1998.76,"endTime":2000.58,"body":"so we have picked our four numbers."},{"startTime":2000.58,"endTime":2003.52,"body":"But then each one occurs, each set occurs,"},{"startTime":2003.52,"endTime":2006.64,"body":"in four factorial, four"},{"startTime":2003.52,"endTime":2006.64,"body":"times two times three times"},{"startTime":2006.64,"endTime":2008.8,"body":"two times one, 24 ways."},{"startTime":2008.8,"endTime":2011.71,"body":"Now, there's a sort of sneaky"},{"startTime":2008.8,"endTime":2011.71,"body":"little trick you can do"},{"startTime":2011.71,"endTime":2013.84,"body":"to write this in a more efficient way"},{"startTime":2013.84,"endTime":2015.137,"body":"with lots of factorials in."},{"startTime":2015.137,"endTime":2016.933,"body":"So I'll just show you that,"},{"startTime":2017.95,"endTime":2019.435,"body":"there are little bits"},{"startTime":2017.95,"endTime":2019.435,"body":"of missing on the end."},{"startTime":2019.435,"endTime":2020.92,"body":"It'll all be there in the end."},{"startTime":2020.92,"endTime":2022.66,"body":"So what I've done, this looks worse,"},{"startTime":2022.66,"endTime":2024.027,"body":"but I promise you, it's better."},{"startTime":2024.027,"endTime":2028.12,"body":"I wish I had the product"},{"startTime":2024.027,"endTime":2028.12,"body":"of all the numbers"},{"startTime":2028.12,"endTime":2029.71,"body":"from one to 10 on the top."},{"startTime":2029.71,"endTime":2031.93,"body":"And so I've made that happen."},{"startTime":2031.93,"endTime":2034.96,"body":"I've forced that to occur"},{"startTime":2031.93,"endTime":2034.96,"body":"by just finishing that list,"},{"startTime":2034.96,"endTime":2036.43,"body":"six, five, four, three, two, one,"},{"startTime":2036.43,"endTime":2038.53,"body":"but then I've put it on the"},{"startTime":2036.43,"endTime":2038.53,"body":"bottom of the fraction as well,"},{"startTime":2038.53,"endTime":2040.756,"body":"so we haven't broken the law. (laughs)"},{"startTime":2040.756,"endTime":2042.61,"body":"These things are equal."},{"startTime":2042.61,"endTime":2044.59,"body":"And now if you do that,"},{"startTime":2044.59,"endTime":2046.673,"body":"you find that what you've actually got"},{"startTime":2046.673,"endTime":2051.16,"body":"is 10 factorial on the top,"},{"startTime":2046.673,"endTime":2051.16,"body":"and then four factorial"},{"startTime":2051.16,"endTime":2052.99,"body":"and six factorial on the bottom."},{"startTime":2052.99,"endTime":2055.24,"body":"So it's kind of a neat"},{"startTime":2052.99,"endTime":2055.24,"body":"nifty way of writing it."},{"startTime":2055.24,"endTime":2057.82,"body":"And you can work it out; it's 210."},{"startTime":2057.82,"endTime":2060.91,"body":"Now, this is a cute little expression,"},{"startTime":2060.91,"endTime":2062.29,"body":"and it works in general."},{"startTime":2062.29,"endTime":2064.9,"body":"So we've done 10 choose four,"},{"startTime":2064.9,"endTime":2067.6,"body":"but the same bit of reasoning works"},{"startTime":2067.6,"endTime":2070.66,"body":"to tell you that the number"},{"startTime":2067.6,"endTime":2070.66,"body":"of ways of choosing R,"},{"startTime":2070.66,"endTime":2073.09,"body":"a set of size R, from a set of size N."},{"startTime":2073.09,"endTime":2074.754,"body":"So R numbers out of N"},{"startTime":2074.754,"endTime":2078.46,"body":"is the same sort of"},{"startTime":2074.754,"endTime":2078.46,"body":"expression with factorials N."},{"startTime":2078.46,"endTime":2081.97,"body":"But I have to accept that's"},{"startTime":2078.46,"endTime":2081.97,"body":"quite a lot of calculating,"},{"startTime":2081.97,"endTime":2083.68,"body":"and multiplying and dividing."},{"startTime":2083.68,"endTime":2085.33,"body":"So I want to show you another context"},{"startTime":2085.33,"endTime":2087.28,"body":"that these numbers arise in,"},{"startTime":2087.28,"endTime":2089.65,"body":"because it gives you an alternative way"},{"startTime":2089.65,"endTime":2091.0,"body":"of finding these numbers"},{"startTime":2091.0,"endTime":2093.25,"body":"that has a lot less"},{"startTime":2091.0,"endTime":2093.25,"body":"multiplying and dividing."},{"startTime":2094.78,"endTime":2098.23,"body":"Okay, so imagine you're"},{"startTime":2094.78,"endTime":2098.23,"body":"in one of those cities"},{"startTime":2098.23,"endTime":2099.85,"body":"that we don't tend to have in Britain,"},{"startTime":2099.85,"endTime":2101.62,"body":"but that actually is"},{"startTime":2099.85,"endTime":2101.62,"body":"kind of well-organized"},{"startTime":2101.62,"endTime":2104.44,"body":"and is laid out on a grid system,"},{"startTime":2104.44,"endTime":2106.69,"body":"a bit like in the middle"},{"startTime":2104.44,"endTime":2106.69,"body":"of New York or something."},{"startTime":2106.69,"endTime":2109.399,"body":"And if you're trying to"},{"startTime":2106.69,"endTime":2109.399,"body":"get from place to place,"},{"startTime":2109.399,"endTime":2111.46,"body":"if I've got a point O and a point P"},{"startTime":2111.46,"endTime":2113.009,"body":"that I'm trying to get from O to P,"},{"startTime":2113.009,"endTime":2114.88,"body":"you're trying to work out a route,"},{"startTime":2114.88,"endTime":2117.22,"body":"well, there might be lots of valid routes,"},{"startTime":2117.22,"endTime":2120.7,"body":"but everything you do is"},{"startTime":2117.22,"endTime":2120.7,"body":"either going right or going up."},{"startTime":2120.7,"endTime":2122.77,"body":"So you're moving through the grid."},{"startTime":2122.77,"endTime":2125.89,"body":"No weird, higgledy-piggledy, curvy lines."},{"startTime":2125.89,"endTime":2130.03,"body":"And so you're only interested"},{"startTime":2125.89,"endTime":2130.03,"body":"in shortest routes,"},{"startTime":2130.03,"endTime":2131.59,"body":"so you're not going to go down or left,"},{"startTime":2131.59,"endTime":2133.45,"body":"or do anything weird like that."},{"startTime":2133.45,"endTime":2136.45,"body":"So to get from O to P, you really,"},{"startTime":2136.45,"endTime":2138.61,"body":"I'll just give one example there."},{"startTime":2138.61,"endTime":2141.91,"body":"Every route will involve"},{"startTime":2138.61,"endTime":2141.91,"body":"going, at some point,"},{"startTime":2141.91,"endTime":2146.44,"body":"right two steps or blocks,"},{"startTime":2141.91,"endTime":2146.44,"body":"and at some point, up three."},{"startTime":2146.44,"endTime":2149.65,"body":"So the total kind of distance"},{"startTime":2146.44,"endTime":2149.65,"body":"you've traveled is five,"},{"startTime":2149.65,"endTime":2151.3,"body":"two right and three up,"},{"startTime":2151.3,"endTime":2153.04,"body":"but you can do it in any order you like."},{"startTime":2153.04,"endTime":2157.09,"body":"So there's one route there"},{"startTime":2153.04,"endTime":2157.09,"body":"that's right up, up right up,"},{"startTime":2157.09,"endTime":2158.14,"body":"but there are many others."},{"startTime":2158.14,"endTime":2160.15,"body":"The question is how many?"},{"startTime":2160.15,"endTime":2162.49,"body":"Well, that's not too difficult to answer,"},{"startTime":2162.49,"endTime":2166.69,"body":"because any route can be"},{"startTime":2162.49,"endTime":2166.69,"body":"specified exactly by saying"},{"startTime":2166.69,"endTime":2170.02,"body":"where, in the five steps"},{"startTime":2166.69,"endTime":2170.02,"body":"that you're going to take,"},{"startTime":2170.02,"endTime":2172.669,"body":"are the two places where you go right?"},{"startTime":2172.669,"endTime":2177.669,"body":"So we're choosing two places,"},{"startTime":2172.669,"endTime":2177.669,"body":"two points out of the five"},{"startTime":2178.33,"endTime":2181.42,"body":"that we're going to make,"},{"startTime":2178.33,"endTime":2181.42,"body":"that's where we go right."},{"startTime":2181.42,"endTime":2182.95,"body":"And so the number of ways to do this"},{"startTime":2182.95,"endTime":2186.79,"body":"is exactly five choose two,"},{"startTime":2182.95,"endTime":2186.79,"body":"choosing two things from five,"},{"startTime":2186.79,"endTime":2190.06,"body":"which is five times four divided"},{"startTime":2186.79,"endTime":2190.06,"body":"by two, which is 10, okay?"},{"startTime":2190.06,"endTime":2192.91,"body":"Very nice, and this works in general."},{"startTime":2192.91,"endTime":2194.83,"body":"So if we're going to any old point P,"},{"startTime":2194.83,"endTime":2198.01,"body":"which involves going, at some"},{"startTime":2194.83,"endTime":2198.01,"body":"point, R steps to the right"},{"startTime":2198.01,"endTime":2201.55,"body":"and U steps up, then the"},{"startTime":2198.01,"endTime":2201.55,"body":"total distance traveled"},{"startTime":2201.55,"endTime":2204.67,"body":"will be R plus U, let's call that N."},{"startTime":2204.67,"endTime":2206.65,"body":"And the number of ways to do it"},{"startTime":2206.65,"endTime":2210.25,"body":"is you just choose which of"},{"startTime":2206.65,"endTime":2210.25,"body":"the places along the route"},{"startTime":2210.25,"endTime":2211.81,"body":"will be where you turn right,"},{"startTime":2211.81,"endTime":2213.49,"body":"where you go a step to the right."},{"startTime":2213.49,"endTime":2216.82,"body":"So you're choosing R"},{"startTime":2213.49,"endTime":2216.82,"body":"things out of a total of N."},{"startTime":2216.82,"endTime":2219.7,"body":"So N choose R; nice."},{"startTime":2219.7,"endTime":2221.08,"body":"So why have I told you this?"},{"startTime":2221.08,"endTime":2224.27,"body":"Because there's a different"},{"startTime":2221.08,"endTime":2224.27,"body":"way of working this out,"},{"startTime":2224.27,"endTime":2225.103,"body":"and it's this."},{"startTime":2225.103,"endTime":2229.27,"body":"To get to that point P,"},{"startTime":2225.103,"endTime":2229.27,"body":"whatever route you take,"},{"startTime":2229.27,"endTime":2232.12,"body":"you'll either, the kind of"},{"startTime":2229.27,"endTime":2232.12,"body":"penultimate place you stop"},{"startTime":2232.12,"endTime":2235.21,"body":"will either be the place just"},{"startTime":2232.12,"endTime":2235.21,"body":"directly to the left of P,"},{"startTime":2235.21,"endTime":2238.39,"body":"or you'll have come from directly below P."},{"startTime":2238.39,"endTime":2240.72,"body":"'Cause to get there, you're"},{"startTime":2238.39,"endTime":2240.72,"body":"only ever going right"},{"startTime":2240.72,"endTime":2241.75,"body":"or you're going up."},{"startTime":2241.75,"endTime":2244.24,"body":"So the last place you have been before P"},{"startTime":2244.24,"endTime":2246.627,"body":"will either be the place"},{"startTime":2244.24,"endTime":2246.627,"body":"directly to the left"},{"startTime":2246.627,"endTime":2248.59,"body":"or the point directly below."},{"startTime":2248.59,"endTime":2250.29,"body":"So the total number of routes to P"},{"startTime":2251.14,"endTime":2253.42,"body":"will be the total number"},{"startTime":2251.14,"endTime":2253.42,"body":"of routes to that point"},{"startTime":2253.42,"endTime":2256.63,"body":"just to the left, plus"},{"startTime":2253.42,"endTime":2256.63,"body":"the total number of points"},{"startTime":2256.63,"endTime":2259.256,"body":"to that place just below."},{"startTime":2259.256,"endTime":2263.65,"body":"And that little argument"},{"startTime":2259.256,"endTime":2263.65,"body":"allows you to build up"},{"startTime":2263.65,"endTime":2266.864,"body":"kind of cumulatively these numbers."},{"startTime":2266.864,"endTime":2270.73,"body":"Anything that is a point"},{"startTime":2266.864,"endTime":2270.73,"body":"directly to the right of O,"},{"startTime":2270.73,"endTime":2272.95,"body":"you get no choice about,"},{"startTime":2270.73,"endTime":2272.95,"body":"'cause there's no up;"},{"startTime":2272.95,"endTime":2275.02,"body":"you just have to go right"},{"startTime":2272.95,"endTime":2275.02,"body":"until you get there."},{"startTime":2275.02,"endTime":2277.21,"body":"So there's always one"},{"startTime":2275.02,"endTime":2277.21,"body":"route to any of those."},{"startTime":2277.21,"endTime":2280.06,"body":"If you're going directly"},{"startTime":2277.21,"endTime":2280.06,"body":"up from O, however far,"},{"startTime":2280.06,"endTime":2282.58,"body":"again, you get no choice,"},{"startTime":2280.06,"endTime":2282.58,"body":"'cause you're never going right,"},{"startTime":2282.58,"endTime":2284.26,"body":"so there's only one route to those."},{"startTime":2284.26,"endTime":2288.04,"body":"But then there's other points"},{"startTime":2284.26,"endTime":2288.04,"body":"you can add your way to."},{"startTime":2288.04,"endTime":2291.4,"body":"So this number two here,"},{"startTime":2288.04,"endTime":2291.4,"body":"we get it by saying"},{"startTime":2291.4,"endTime":2293.959,"body":"you've either come from below,"},{"startTime":2291.4,"endTime":2293.959,"body":"there's one way to get there,"},{"startTime":2293.959,"endTime":2296.59,"body":"or you've come from the left,"},{"startTime":2293.959,"endTime":2296.59,"body":"there's one way to get there."},{"startTime":2296.59,"endTime":2298.96,"body":"And then one plus one is two, right?"},{"startTime":2298.96,"endTime":2299.86,"body":"Two plus one is three."},{"startTime":2299.86,"endTime":2300.693,"body":"And you can build up and up."},{"startTime":2300.693,"endTime":2302.68,"body":"And by the time you get to that 10 there,"},{"startTime":2302.68,"endTime":2304.18,"body":"you're adding four and six."},{"startTime":2304.18,"endTime":2306.88,"body":"So you can build these"},{"startTime":2304.18,"endTime":2306.88,"body":"things up just by adding,"},{"startTime":2306.88,"endTime":2309.28,"body":"which feels nicer"},{"startTime":2309.28,"endTime":2311.463,"body":"than doing all that"},{"startTime":2309.28,"endTime":2311.463,"body":"multiplying and dividing."},{"startTime":2312.61,"endTime":2315.73,"body":"So you can carry on the"},{"startTime":2312.61,"endTime":2315.73,"body":"grid as far as you like."},{"startTime":2315.73,"endTime":2317.44,"body":"I'm just going to throw"},{"startTime":2315.73,"endTime":2317.44,"body":"away the furniture there"},{"startTime":2317.44,"endTime":2318.55,"body":"and leave that grid."},{"startTime":2318.55,"endTime":2321.49,"body":"Now I'm going to rotate"},{"startTime":2318.55,"endTime":2321.49,"body":"it now and just show you,"},{"startTime":2321.49,"endTime":2326.037,"body":"because perhaps some of us"},{"startTime":2321.49,"endTime":2326.037,"body":"have seen this triangle before."},{"startTime":2327.58,"endTime":2329.38,"body":"It's exactly the same thing;"},{"startTime":2329.38,"endTime":2331.21,"body":"I've just rotated it round a bit."},{"startTime":2331.21,"endTime":2333.01,"body":"And in this triangle on the left now,"},{"startTime":2333.01,"endTime":2336.82,"body":"every number is made by adding"},{"startTime":2333.01,"endTime":2336.82,"body":"the two immediately above it."},{"startTime":2336.82,"endTime":2339.13,"body":"Or if there isn't something on one side,"},{"startTime":2339.13,"endTime":2341.08,"body":"that's an invisible zero, okay?"},{"startTime":2341.08,"endTime":2345.28,"body":"Right, so we can work out"},{"startTime":2341.08,"endTime":2345.28,"body":"these N choose R things"},{"startTime":2345.28,"endTime":2347.02,"body":"by using a triangle like this."},{"startTime":2347.02,"endTime":2347.853,"body":"And we can write it on a piece of paper"},{"startTime":2347.853,"endTime":2350.71,"body":"and carry it in our pocket if we want to."},{"startTime":2350.71,"endTime":2353.683,"body":"Now, to cycle back to"},{"startTime":2350.71,"endTime":2353.683,"body":"something we mentioned before,"},{"startTime":2355.45,"endTime":2358.21,"body":"this helps us with multiplying"},{"startTime":2355.45,"endTime":2358.21,"body":"our brackets as well."},{"startTime":2358.21,"endTime":2359.8,"body":"If you're doing something like one plus X"},{"startTime":2359.8,"endTime":2361.42,"body":"all to the power of five,"},{"startTime":2361.42,"endTime":2363.82,"body":"that's imprinted in those brackets,"},{"startTime":2363.82,"endTime":2366.22,"body":"what's the coefficient of X squared, say?"},{"startTime":2366.22,"endTime":2367.3,"body":"Well, when you're multiplying this out,"},{"startTime":2367.3,"endTime":2369.19,"body":"how will you get X squared?"},{"startTime":2369.19,"endTime":2372.269,"body":"Well, you're going to have to choose an X"},{"startTime":2372.269,"endTime":2375.25,"body":"from exactly two of the brackets."},{"startTime":2375.25,"endTime":2379.93,"body":"So there are five choose two"},{"startTime":2375.25,"endTime":2379.93,"body":"ways to do that, which is 10."},{"startTime":2379.93,"endTime":2384.19,"body":"Okay, so we can, this, this"},{"startTime":2379.93,"endTime":2384.19,"body":"triangle in these coefficients"},{"startTime":2384.19,"endTime":2387.4,"body":"are exactly the binomial"},{"startTime":2384.19,"endTime":2387.4,"body":"coefficients I alluded to earlier"},{"startTime":2387.4,"endTime":2388.72,"body":"but didn't specify."},{"startTime":2388.72,"endTime":2390.484,"body":"So they are really, really useful"},{"startTime":2390.484,"endTime":2392.83,"body":"not just for lotteries"},{"startTime":2390.484,"endTime":2392.83,"body":"but also in working out"},{"startTime":2392.83,"endTime":2395.977,"body":"these other probabilistic"},{"startTime":2392.83,"endTime":2395.977,"body":"calculations we saw before."},{"startTime":2395.977,"endTime":2400.48,"body":"And this triangle, Pascal"},{"startTime":2395.977,"endTime":2400.48,"body":"wrote a treatise on it,"},{"startTime":2400.48,"endTime":2403.12,"body":"and he wrote about the"},{"startTime":2400.48,"endTime":2403.12,"body":"binomial coefficients,"},{"startTime":2403.12,"endTime":2403.953,"body":"and it's because of that"},{"startTime":2403.953,"endTime":2406.93,"body":"that we now call this"},{"startTime":2403.953,"endTime":2406.93,"body":"triangle \"Pascal's triangle.\""},{"startTime":2406.93,"endTime":2410.774,"body":"I have to say it was"},{"startTime":2406.93,"endTime":2410.774,"body":"known to Indian, Chinese,"},{"startTime":2410.774,"endTime":2414.37,"body":"Arabic and Persian"},{"startTime":2410.774,"endTime":2414.37,"body":"mathematicians before Pascal,"},{"startTime":2414.37,"endTime":2417.51,"body":"but this was how kind of,"},{"startTime":2414.37,"endTime":2417.51,"body":"I guess we heard about it"},{"startTime":2417.51,"endTime":2421.06,"body":"in the UK from Pascal, so we"},{"startTime":2417.51,"endTime":2421.06,"body":"call it \"Pascal's triangle.\""},{"startTime":2421.06,"endTime":2423.643,"body":"Okay, we now can work out lottery odds."},{"startTime":2424.54,"endTime":2425.863,"body":"In the UK Lotto,"},{"startTime":2426.73,"endTime":2430.9,"body":"you choose six balls"},{"startTime":2426.73,"endTime":2430.9,"body":"from six numbers from 59."},{"startTime":2430.9,"endTime":2435.9,"body":"And so the number of ways"},{"startTime":2430.9,"endTime":2435.9,"body":"to do that is 59 choose six,"},{"startTime":2436.21,"endTime":2439.27,"body":"which we can work out with a"},{"startTime":2436.21,"endTime":2439.27,"body":"actually pretty big triangle,"},{"startTime":2439.27,"endTime":2441.258,"body":"right, or we can use our calculation"},{"startTime":2441.258,"endTime":2443.53,"body":"for binomial coefficients."},{"startTime":2443.53,"endTime":2445.814,"body":"And we can work it out that"},{"startTime":2443.53,"endTime":2445.814,"body":"the odds of doing this,"},{"startTime":2445.814,"endTime":2450.1,"body":"so there are 45 million-ish,"},{"startTime":2445.814,"endTime":2450.1,"body":"that's 45 million-ish,"},{"startTime":2450.1,"endTime":2453.13,"body":"59 choose six is about 45 million."},{"startTime":2453.13,"endTime":2456.76,"body":"So the odds of us happening"},{"startTime":2453.13,"endTime":2456.76,"body":"upon the actual correct numbers,"},{"startTime":2456.76,"endTime":2460.15,"body":"the jackpot-winning numbers,"},{"startTime":2456.76,"endTime":2460.15,"body":"are one in 45 million."},{"startTime":2460.15,"endTime":2462.22,"body":"Not great (laughs) odds."},{"startTime":2463.66,"endTime":2466.6,"body":"Okay, there are other ways"},{"startTime":2463.66,"endTime":2466.6,"body":"you can win some money."},{"startTime":2466.6,"endTime":2471.16,"body":"So if you match exactly three"},{"startTime":2466.6,"endTime":2471.16,"body":"numbers, you win 30 pounds."},{"startTime":2471.16,"endTime":2472.51,"body":"Well, that's all right."},{"startTime":2472.51,"endTime":2476.59,"body":"So we can do the math to see"},{"startTime":2472.51,"endTime":2476.59,"body":"what are the odds of that."},{"startTime":2476.59,"endTime":2479.2,"body":"If you're matching exactly three numbers,"},{"startTime":2479.2,"endTime":2480.55,"body":"then three of your numbers"},{"startTime":2480.55,"endTime":2482.83,"body":"must be chosen from the"},{"startTime":2480.55,"endTime":2482.83,"body":"six winning numbers."},{"startTime":2482.83,"endTime":2485.29,"body":"So that's six choose three."},{"startTime":2485.29,"endTime":2487.48,"body":"But then the other numbers, three numbers,"},{"startTime":2487.48,"endTime":2491.71,"body":"have to be coming from the"},{"startTime":2487.48,"endTime":2491.71,"body":"53 not-winning numbers."},{"startTime":2491.71,"endTime":2493.99,"body":"So that's 53 choose three."},{"startTime":2493.99,"endTime":2495.876,"body":"And if you multiply those together,"},{"startTime":2495.876,"endTime":2500.71,"body":"you get that there are about"},{"startTime":2495.876,"endTime":2500.71,"body":"468,000 ways of buying a ticket"},{"startTime":2500.71,"endTime":2502.72,"body":"that matches exactly three numbers"},{"startTime":2502.72,"endTime":2507.22,"body":"out of a total of 45"},{"startTime":2502.72,"endTime":2507.22,"body":"million possible tickets."},{"startTime":2507.22,"endTime":2508.45,"body":"So if you work that out,"},{"startTime":2508.45,"endTime":2512.02,"body":"it's about a one in 96"},{"startTime":2508.45,"endTime":2512.02,"body":"chance of winning 30 quid,"},{"startTime":2512.02,"endTime":2515.311,"body":"which again, hmm, (laughs)"},{"startTime":2512.02,"endTime":2515.311,"body":"doesn't seem amazing."},{"startTime":2515.311,"endTime":2518.59,"body":"Okay, but you know, I've got you in here"},{"startTime":2518.59,"endTime":2521.71,"body":"with a promise that you"},{"startTime":2518.59,"endTime":2521.71,"body":"can win the lottery, right?"},{"startTime":2521.71,"endTime":2523.18,"body":"Well, sort of. (laughs)"},{"startTime":2523.18,"endTime":2524.26,"body":"Okay, so how do you do it?"},{"startTime":2524.26,"endTime":2526.36,"body":"It's easy; it's so easy."},{"startTime":2526.36,"endTime":2528.61,"body":"What you do is you watch the lottery,"},{"startTime":2528.61,"endTime":2529.627,"body":"you write down the numbers,"},{"startTime":2529.627,"endTime":2532.51,"body":"and then you quickly"},{"startTime":2529.627,"endTime":2532.51,"body":"invent the time machine,"},{"startTime":2532.51,"endTime":2536.2,"body":"and go back and buy a"},{"startTime":2532.51,"endTime":2536.2,"body":"winning, no? (laughs)"},{"startTime":2536.2,"endTime":2538.24,"body":"Okay, so, okay, I jest."},{"startTime":2538.24,"endTime":2540.07,"body":"I do remember there was a"},{"startTime":2538.24,"endTime":2540.07,"body":"program a few years ago,"},{"startTime":2540.07,"endTime":2541.33,"body":"a science fiction program,"},{"startTime":2541.33,"endTime":2543.31,"body":"where they decided they"},{"startTime":2541.33,"endTime":2543.31,"body":"were going to do this."},{"startTime":2543.31,"endTime":2545.8,"body":"Somehow, even though they'd"},{"startTime":2543.31,"endTime":2545.8,"body":"invented a time machine,"},{"startTime":2545.8,"endTime":2547.93,"body":"they couldn't get their"},{"startTime":2545.8,"endTime":2547.93,"body":"grant funding renewed,"},{"startTime":2547.93,"endTime":2551.38,"body":"which, (laughs) well,"},{"startTime":2547.93,"endTime":2551.38,"body":"it's difficult out there."},{"startTime":2551.38,"endTime":2553.12,"body":"So what they did, they got the numbers,"},{"startTime":2553.12,"endTime":2554.11,"body":"then they somehow found"},{"startTime":2554.11,"endTime":2556.15,"body":"that they'd only matched four numbers."},{"startTime":2556.15,"endTime":2557.68,"body":"But this is a cute little plot twist,"},{"startTime":2557.68,"endTime":2559.06,"body":"and you might like to investigate"},{"startTime":2559.06,"endTime":2561.91,"body":"which sets of six numbers this works for."},{"startTime":2561.91,"endTime":2563.95,"body":"When they got to the"},{"startTime":2561.91,"endTime":2563.95,"body":"place to buy their ticket,"},{"startTime":2563.95,"endTime":2566.92,"body":"they looked at the bit"},{"startTime":2563.95,"endTime":2566.92,"body":"of paper upside down."},{"startTime":2566.92,"endTime":2570.31,"body":"And so they found they'd"},{"startTime":2566.92,"endTime":2570.31,"body":"only matched four numbers."},{"startTime":2570.31,"endTime":2572.35,"body":"I don't know if these were"},{"startTime":2570.31,"endTime":2572.35,"body":"the exact sets, but anyway,"},{"startTime":2572.35,"endTime":2574.045,"body":"you can work out all the possibilities."},{"startTime":2574.045,"endTime":2577.09,"body":"Okay, this isn't my real advice."},{"startTime":2577.09,"endTime":2578.62,"body":"You don't have to build a time machine."},{"startTime":2578.62,"endTime":2580.63,"body":"I'm going to present you with six ideas"},{"startTime":2580.63,"endTime":2582.67,"body":"to finish off with six ways"},{"startTime":2582.67,"endTime":2585.43,"body":"to maximize your lottery"},{"startTime":2582.67,"endTime":2585.43,"body":"winnings, none of which, bonus,"},{"startTime":2585.43,"endTime":2588.37,"body":"none of which require you to"},{"startTime":2585.43,"endTime":2588.37,"body":"break the laws of physics."},{"startTime":2588.37,"endTime":2592.72,"body":"Okay, so the kind of the game"},{"startTime":2588.37,"endTime":2592.72,"body":"is, you've got some budget,"},{"startTime":2592.72,"endTime":2594.04,"body":"maybe you've got two pounds a week,"},{"startTime":2594.04,"endTime":2596.11,"body":"that's how much a lottery"},{"startTime":2594.04,"endTime":2596.11,"body":"ticket costs at the moment,"},{"startTime":2596.11,"endTime":2600.49,"body":"you've got some budget that"},{"startTime":2596.11,"endTime":2600.49,"body":"you can invest in the lottery."},{"startTime":2600.49,"endTime":2601.81,"body":"What is the way to end the year"},{"startTime":2601.81,"endTime":2604.33,"body":"with the biggest amount of money?"},{"startTime":2604.33,"endTime":2607.053,"body":"Tip number one, do not play the lottery."},{"startTime":2607.053,"endTime":2608.29,"body":"(students laugh)"},{"startTime":2608.29,"endTime":2610.45,"body":"Okay, I'm going to have"},{"startTime":2608.29,"endTime":2610.45,"body":"to say this, right?"},{"startTime":2610.45,"endTime":2611.41,"body":"If you don't buy a ticket,"},{"startTime":2611.41,"endTime":2612.76,"body":"instead you put that money in the bank,"},{"startTime":2612.76,"endTime":2615.49,"body":"at the end of the year,"},{"startTime":2612.76,"endTime":2615.49,"body":"you will have 104 pounds,"},{"startTime":2615.49,"endTime":2620.05,"body":"and your lottery-playing alter"},{"startTime":2615.49,"endTime":2620.05,"body":"ego is likely to have less."},{"startTime":2620.05,"endTime":2624.19,"body":"The amount of ticket revenue"},{"startTime":2620.05,"endTime":2624.19,"body":"that is given out in prizes"},{"startTime":2624.19,"endTime":2627.7,"body":"is higher than I thought,"},{"startTime":2624.19,"endTime":2627.7,"body":"actually; it's just over 50%."},{"startTime":2627.7,"endTime":2628.9,"body":"So, of course, that includes"},{"startTime":2628.9,"endTime":2630.809,"body":"the jackpot winnings and everything."},{"startTime":2630.809,"endTime":2635.809,"body":"Your, over the very long-term,"},{"startTime":2630.809,"endTime":2635.809,"body":"the expected winnings are,"},{"startTime":2636.52,"endTime":2638.26,"body":"well, certainly not 100%."},{"startTime":2638.26,"endTime":2641.92,"body":"Your lottery-playing alter"},{"startTime":2638.26,"endTime":2641.92,"body":"ego will have, on average,"},{"startTime":2641.92,"endTime":2644.65,"body":"at the end of the year, about 50 quid,"},{"startTime":2644.65,"endTime":2646.21,"body":"and you'll have 104 quid."},{"startTime":2646.21,"endTime":2648.843,"body":"So, you know, this is the way to do it."},{"startTime":2650.23,"endTime":2653.14,"body":"Of course, if it's fun, that's fine."},{"startTime":2653.14,"endTime":2656.05,"body":"I think we just have to"},{"startTime":2653.14,"endTime":2656.05,"body":"remember what we're paying for"},{"startTime":2656.05,"endTime":2658.57,"body":"is not the definite"},{"startTime":2656.05,"endTime":2658.57,"body":"likelihood of getting rich."},{"startTime":2658.57,"endTime":2660.34,"body":"It's that little free sigh of excitement"},{"startTime":2660.34,"endTime":2662.05,"body":"of \"maybe I'll do this,\" or maybe you like"},{"startTime":2662.05,"endTime":2664.0,"body":"that you're giving some"},{"startTime":2662.05,"endTime":2664.0,"body":"money to good causes,"},{"startTime":2664.0,"endTime":2665.236,"body":"or something like this."},{"startTime":2665.236,"endTime":2668.95,"body":"Never, never bet more than"},{"startTime":2665.236,"endTime":2668.95,"body":"you can cheerfully lose,"},{"startTime":2668.95,"endTime":2671.02,"body":"and be happy and have a"},{"startTime":2668.95,"endTime":2671.02,"body":"smile on your face, okay?"},{"startTime":2671.02,"endTime":2672.49,"body":"So that's our tip number one."},{"startTime":2672.49,"endTime":2674.56,"body":"If you must play the lottery,"},{"startTime":2674.56,"endTime":2676.12,"body":"you could play a different lottery."},{"startTime":2676.12,"endTime":2679.24,"body":"So there are lots of"},{"startTime":2676.12,"endTime":2679.24,"body":"lotteries around the world."},{"startTime":2679.24,"endTime":2681.44,"body":"Famously, the U.S. Powerball lottery"},{"startTime":2683.23,"endTime":2684.46,"body":"tends to have very big prizes."},{"startTime":2684.46,"endTime":2688.943,"body":"The reason is the chance of"},{"startTime":2684.46,"endTime":2688.943,"body":"jackpot is one in 292 million."},{"startTime":2690.91,"endTime":2694.21,"body":"That's quite hard to get, so"},{"startTime":2690.91,"endTime":2694.21,"body":"you quite often get rollovers."},{"startTime":2694.21,"endTime":2697.78,"body":"In 2022, there were 40"},{"startTime":2694.21,"endTime":2697.78,"body":"draws without a win,"},{"startTime":2697.78,"endTime":2701.8,"body":"and the rollover amount"},{"startTime":2697.78,"endTime":2701.8,"body":"reached $2 billion, (laughs)"},{"startTime":2701.8,"endTime":2702.9,"body":"which is a lot, a lot."},{"startTime":2704.05,"endTime":2707.68,"body":"The easiest probably one to"},{"startTime":2704.05,"endTime":2707.68,"body":"win is the Polish Mini Lotto,"},{"startTime":2707.68,"endTime":2711.49,"body":"chance of about one in 850,000,"},{"startTime":2711.49,"endTime":2713.59,"body":"'cause you're drawing only five balls,"},{"startTime":2713.59,"endTime":2715.81,"body":"and you're drawing them from"},{"startTime":2713.59,"endTime":2715.81,"body":"only the numbers one to 42."},{"startTime":2715.81,"endTime":2716.643,"body":"So you can sort of imagine"},{"startTime":2716.643,"endTime":2719.97,"body":"that that's going to be a"},{"startTime":2716.643,"endTime":2719.97,"body":"lower number of possibilities."},{"startTime":2719.97,"endTime":2722.35,"body":"On the other extreme, Italy's Lotto,"},{"startTime":2722.35,"endTime":2725.86,"body":"one in 622 million chance."},{"startTime":2725.86,"endTime":2728.71,"body":"So again, that's very hard to do."},{"startTime":2728.71,"endTime":2732.13,"body":"Now, you might think, right, there are,"},{"startTime":2732.13,"endTime":2733.63,"body":"in the American lottery,"},{"startTime":2733.63,"endTime":2738.343,"body":"292-ish million possible"},{"startTime":2733.63,"endTime":2738.343,"body":"tickets you could buy."},{"startTime":2739.194,"endTime":2744.194,"body":"If the winnings reach $2"},{"startTime":2739.194,"endTime":2744.194,"body":"billion, it's $2 per ticket,"},{"startTime":2747.73,"endTime":2751.36,"body":"should I spend $584 million"},{"startTime":2747.73,"endTime":2751.36,"body":"to get all possible tickets?"},{"startTime":2751.36,"endTime":2754.09,"body":"I can guarantee I will have"},{"startTime":2751.36,"endTime":2754.09,"body":"a winning ticket in there."},{"startTime":2754.09,"endTime":2757.57,"body":"Okay, this is, no, (laughs)"},{"startTime":2754.09,"endTime":2757.57,"body":"of course I'm going to say no."},{"startTime":2757.57,"endTime":2759.94,"body":"The reason is, what do you"},{"startTime":2757.57,"endTime":2759.94,"body":"think everyone else is doing"},{"startTime":2759.94,"endTime":2762.31,"body":"when there's a Powerball multiple rollover"},{"startTime":2762.31,"endTime":2763.96,"body":"and the prize is $2 billion?"},{"startTime":2763.96,"endTime":2765.31,"body":"Everyone's buying a ticket."},{"startTime":2765.31,"endTime":2769.12,"body":"Your chances of having"},{"startTime":2765.31,"endTime":2769.12,"body":"to share the jackpot"},{"startTime":2769.12,"endTime":2770.8,"body":"are really quite high."},{"startTime":2770.8,"endTime":2772.9,"body":"And even, I mean, also you have to pay tax"},{"startTime":2772.9,"endTime":2774.07,"body":"on this 2 billion, right?"},{"startTime":2774.07,"endTime":2776.14,"body":"So once you've done that,"},{"startTime":2776.14,"endTime":2778.75,"body":"if you have to share it with"},{"startTime":2776.14,"endTime":2778.75,"body":"even one or two other people,"},{"startTime":2778.75,"endTime":2780.94,"body":"you are not making money;"},{"startTime":2778.75,"endTime":2780.94,"body":"you're making a loss."},{"startTime":2780.94,"endTime":2782.41,"body":"So that's really quite risky."},{"startTime":2782.41,"endTime":2785.827,"body":"This is why hedge fund managers"},{"startTime":2782.41,"endTime":2785.827,"body":"are not doing this, right?"},{"startTime":2785.827,"endTime":2787.63,"body":"'Cause it's very risky strategy."},{"startTime":2787.63,"endTime":2789.52,"body":"You're probably going to have to share."},{"startTime":2789.52,"endTime":2794.26,"body":"There was one instance where"},{"startTime":2789.52,"endTime":2794.26,"body":"a group of people realized,"},{"startTime":2794.26,"endTime":2797.02,"body":"this is in 1992 in the Irish Lotto,"},{"startTime":2797.02,"endTime":2800.83,"body":"which was easier to win, only"},{"startTime":2797.02,"endTime":2800.83,"body":"1.9 million combinations;"},{"startTime":2800.83,"endTime":2802.518,"body":"tickets were 50 p."},{"startTime":2802.518,"endTime":2807.518,"body":"There was a rollover, and the"},{"startTime":2802.518,"endTime":2807.518,"body":"amount reached 1.7 million,"},{"startTime":2807.979,"endTime":2811.06,"body":"and to buy one of each ticket"},{"startTime":2807.979,"endTime":2811.06,"body":"was less than a million."},{"startTime":2811.06,"endTime":2812.893,"body":"And some people tried to do this."},{"startTime":2813.76,"endTime":2815.78,"body":"They did manage to get a winning ticket."},{"startTime":2815.78,"endTime":2818.32,"body":"Well, they managed to get an estimated 80%"},{"startTime":2818.32,"endTime":2820.45,"body":"of all possible tickets,"},{"startTime":2818.32,"endTime":2820.45,"body":"which is pretty good,"},{"startTime":2820.45,"endTime":2821.89,"body":"since this was a time when, like,"},{"startTime":2821.89,"endTime":2823.33,"body":"you couldn't just buy things online."},{"startTime":2823.33,"endTime":2825.91,"body":"You had to physically go"},{"startTime":2823.33,"endTime":2825.91,"body":"to shops and buy tickets"},{"startTime":2825.91,"endTime":2827.77,"body":"one at a time from terminals, right?"},{"startTime":2827.77,"endTime":2830.65,"body":"So they, somehow, they managed"},{"startTime":2827.77,"endTime":2830.65,"body":"to get 80% of the tickets"},{"startTime":2830.65,"endTime":2831.67,"body":"over the course of a week."},{"startTime":2831.67,"endTime":2834.76,"body":"They did have a winning ticket,"},{"startTime":2831.67,"endTime":2834.76,"body":"but they had to share it."},{"startTime":2834.76,"endTime":2838.03,"body":"Now, they won some"},{"startTime":2834.76,"endTime":2838.03,"body":"subsidiary prizes as well."},{"startTime":2838.03,"endTime":2840.16,"body":"So they've never been quite clear"},{"startTime":2840.16,"endTime":2842.98,"body":"about exactly how much"},{"startTime":2840.16,"endTime":2842.98,"body":"they won in the end,"},{"startTime":2842.98,"endTime":2844.21,"body":"it was a group of people,"},{"startTime":2844.21,"endTime":2847.06,"body":"but they may have won a couple"},{"startTime":2844.21,"endTime":2847.06,"body":"of hundred thousand pounds."},{"startTime":2847.06,"endTime":2849.22,"body":"They were not millionaires for life."},{"startTime":2849.22,"endTime":2850.633,"body":"But it has been done."},{"startTime":2851.47,"endTime":2853.99,"body":"Okay, number three, join a syndicate."},{"startTime":2853.99,"endTime":2858.765,"body":"So about a fifth of the"},{"startTime":2853.99,"endTime":2858.765,"body":"jackpots are won by syndicates."},{"startTime":2858.765,"endTime":2860.2,"body":"If you're in a syndicate,"},{"startTime":2860.2,"endTime":2862.513,"body":"you're kind of spreading your risk a bit."},{"startTime":2863.5,"endTime":2865.6,"body":"You each put in the"},{"startTime":2863.5,"endTime":2865.6,"body":"price of one ticket, say,"},{"startTime":2865.6,"endTime":2868.81,"body":"but then you're actually"},{"startTime":2865.6,"endTime":2868.81,"body":"buying a share of 10 tickets."},{"startTime":2868.81,"endTime":2872.53,"body":"And so the chances, you're"},{"startTime":2868.81,"endTime":2872.53,"body":"making a little bit higher chance"},{"startTime":2872.53,"endTime":2875.2,"body":"for yourself of having"},{"startTime":2872.53,"endTime":2875.2,"body":"a share of a jackpot,"},{"startTime":2875.2,"endTime":2876.61,"body":"but it's a share."},{"startTime":2876.61,"endTime":2878.56,"body":"It's a share of a jackpot."},{"startTime":2878.56,"endTime":2880.233,"body":"So you know, that's factored in, right?"},{"startTime":2880.233,"endTime":2881.95,"body":"You're never going to be like the one"},{"startTime":2881.95,"endTime":2884.35,"body":"who wins 10 million pounds;"},{"startTime":2881.95,"endTime":2884.35,"body":"you'll be sharing that."},{"startTime":2884.35,"endTime":2886.78,"body":"So since it swings,"},{"startTime":2884.35,"endTime":2886.78,"body":"it's a roundabout thing."},{"startTime":2886.78,"endTime":2890.47,"body":"I did wonder, though, if you're"},{"startTime":2886.78,"endTime":2890.47,"body":"buying a bunch of tickets,"},{"startTime":2890.47,"endTime":2895.47,"body":"maybe there's a way to guarantee"},{"startTime":2890.47,"endTime":2895.47,"body":"at least getting any prize."},{"startTime":2895.51,"endTime":2897.19,"body":"How many tickets do you have to buy"},{"startTime":2897.19,"endTime":2899.799,"body":"in order to guarantee"},{"startTime":2897.19,"endTime":2899.799,"body":"matching three numbers"},{"startTime":2899.799,"endTime":2902.053,"body":"if you're careful about the combinations?"},{"startTime":2903.31,"endTime":2905.803,"body":"Interestingly, I'll just"},{"startTime":2903.31,"endTime":2905.803,"body":"look up the formula,"},{"startTime":2906.7,"endTime":2909.91,"body":"unsolved form of mathematics,"},{"startTime":2906.7,"endTime":2909.91,"body":"the Lotto design problem."},{"startTime":2909.91,"endTime":2911.71,"body":"If you can work out this minimum number,"},{"startTime":2911.71,"endTime":2914.95,"body":"then you've got a good math"},{"startTime":2911.71,"endTime":2914.95,"body":"research career ahead of you."},{"startTime":2914.95,"endTime":2916.9,"body":"Nobody has got a formula"},{"startTime":2914.95,"endTime":2916.9,"body":"for this, in general."},{"startTime":2916.9,"endTime":2918.61,"body":"I mean, there's some little examples."},{"startTime":2918.61,"endTime":2921.64,"body":"So this number is called L."},{"startTime":2921.64,"endTime":2925.21,"body":"If you want to be sure"},{"startTime":2921.64,"endTime":2925.21,"body":"of matching one number,"},{"startTime":2925.21,"endTime":2928.18,"body":"then if you buy 10"},{"startTime":2925.21,"endTime":2928.18,"body":"tickets, you can do that,"},{"startTime":2928.18,"endTime":2930.91,"body":"because there are 59"},{"startTime":2928.18,"endTime":2930.91,"body":"numbers, six on each ticket."},{"startTime":2930.91,"endTime":2932.02,"body":"So 10 tickets will mean"},{"startTime":2932.02,"endTime":2934.63,"body":"you can have one of"},{"startTime":2932.02,"endTime":2934.63,"body":"each number represented."},{"startTime":2934.63,"endTime":2936.58,"body":"But, of course, you don't"},{"startTime":2934.63,"endTime":2936.58,"body":"get a prize for that."},{"startTime":2936.58,"endTime":2939.37,"body":"There are some theorems"},{"startTime":2936.58,"endTime":2939.37,"body":"that tell you bounds."},{"startTime":2939.37,"endTime":2941.86,"body":"So it's known that if you buy, well,"},{"startTime":2941.86,"endTime":2944.41,"body":"you need to buy at least 22 tickets"},{"startTime":2944.41,"endTime":2948.46,"body":"in order to have a hope of"},{"startTime":2944.41,"endTime":2948.46,"body":"matching two numbers guaranteed."},{"startTime":2948.46,"endTime":2951.4,"body":"But the smallest prize"},{"startTime":2948.46,"endTime":2951.4,"body":"is for three numbers."},{"startTime":2951.4,"endTime":2954.73,"body":"And even 22 tickets is already 44 pounds,"},{"startTime":2954.73,"endTime":2956.44,"body":"and the prize for three"},{"startTime":2954.73,"endTime":2956.44,"body":"numbers is 30 pounds."},{"startTime":2956.44,"endTime":2959.497,"body":"So again, (laughs) that's"},{"startTime":2956.44,"endTime":2959.497,"body":"not going to help you."},{"startTime":2959.497,"endTime":2962.99,"body":"There are some specific"},{"startTime":2959.497,"endTime":2962.99,"body":"cases where maybe it can help"},{"startTime":2962.99,"endTime":2965.23,"body":"in unusual rollover type situations."},{"startTime":2965.23,"endTime":2967.243,"body":"But usually that's not going to be good."},{"startTime":2968.08,"endTime":2971.417,"body":"Okay, tick four of six, never"},{"startTime":2968.08,"endTime":2971.417,"body":"buy your ticket on a Monday."},{"startTime":2971.417,"endTime":2974.8,"body":"This is a UK-specific tip, by the way."},{"startTime":2974.8,"endTime":2977.8,"body":"And the reason is that in the UK,"},{"startTime":2977.8,"endTime":2980.65,"body":"Lotto draws are on"},{"startTime":2977.8,"endTime":2980.65,"body":"Wednesdays and Saturdays."},{"startTime":2980.65,"endTime":2982.78,"body":"So a bit macabre, right,"},{"startTime":2982.78,"endTime":2986.71,"body":"but the chance of winning the"},{"startTime":2982.78,"endTime":2986.71,"body":"jackpot is one in 45 million,"},{"startTime":2986.71,"endTime":2987.94,"body":"as we've seen."},{"startTime":2987.94,"endTime":2990.1,"body":"Unfortunately, anyone's chance"},{"startTime":2990.1,"endTime":2993.1,"body":"of dying in any 24-hour period"},{"startTime":2993.1,"endTime":2996.343,"body":"is higher than one in 45 million, so"},{"startTime":2996.343,"endTime":2997.508,"body":"(students laugh)"},{"startTime":2997.508,"endTime":3000.72,"body":"you have a higher chance of dying"},{"startTime":3000.72,"endTime":3004.02,"body":"before you find that"},{"startTime":3000.72,"endTime":3004.02,"body":"you've won the lottery"},{"startTime":3004.02,"endTime":3005.97,"body":"if you buy your ticket"},{"startTime":3004.02,"endTime":3005.97,"body":"too far ahead of time."},{"startTime":3005.97,"endTime":3008.16,"body":"And the extreme of this,"},{"startTime":3008.16,"endTime":3010.71,"body":"I looked up some accident statistics,"},{"startTime":3010.71,"endTime":3013.53,"body":"you have a one in 30 million chance,"},{"startTime":3013.53,"endTime":3015.54,"body":"so more likely than winning the jackpot,"},{"startTime":3015.54,"endTime":3020.25,"body":"one in 30 million chance"},{"startTime":3015.54,"endTime":3020.25,"body":"of dying from falling over"},{"startTime":3020.25,"endTime":3023.563,"body":"while trying to put your"},{"startTime":3020.25,"endTime":3023.563,"body":"trousers on. (laughs)"},{"startTime":3025.47,"endTime":3028.617,"body":"So just, okay, be"},{"startTime":3025.47,"endTime":3028.617,"body":"careful out there, right?"},{"startTime":3028.617,"endTime":3029.45,"body":"(students laugh)"},{"startTime":3029.45,"endTime":3033.09,"body":"And if you must gamble, do"},{"startTime":3029.45,"endTime":3033.09,"body":"it at the last minute, right?"},{"startTime":3033.09,"endTime":3037.62,"body":"So point five, don't try and"},{"startTime":3033.09,"endTime":3037.62,"body":"prove a point about randomness."},{"startTime":3037.62,"endTime":3040.2,"body":"So as we've discussed with coin tossing,"},{"startTime":3040.2,"endTime":3044.16,"body":"any particular set of six numbers"},{"startTime":3044.16,"endTime":3048.36,"body":"is just as likely to come up"},{"startTime":3044.16,"endTime":3048.36,"body":"as any other particular set,"},{"startTime":3048.36,"endTime":3050.94,"body":"whether or not we attach"},{"startTime":3048.36,"endTime":3050.94,"body":"any specific significance"},{"startTime":3050.94,"endTime":3053.49,"body":"to them ourselves as humans."},{"startTime":3053.49,"endTime":3055.86,"body":"So one, two, three, four, five, six"},{"startTime":3055.86,"endTime":3059.34,"body":"has a one in 45 million"},{"startTime":3055.86,"endTime":3059.34,"body":"chance of coming up."},{"startTime":3059.34,"endTime":3061.17,"body":"It has exactly the same"},{"startTime":3059.34,"endTime":3061.17,"body":"chance of coming up"},{"startTime":3061.17,"endTime":3065.13,"body":"as this sequence of numbers,"},{"startTime":3061.17,"endTime":3065.13,"body":"or any other specific sequence."},{"startTime":3065.13,"endTime":3066.63,"body":"They've all got the same"},{"startTime":3065.13,"endTime":3066.63,"body":"chance of coming up."},{"startTime":3066.63,"endTime":3068.94,"body":"So what we can do, all we can really do"},{"startTime":3068.94,"endTime":3071.01,"body":"is to try and make sure"},{"startTime":3071.01,"endTime":3074.34,"body":"we don't have to share"},{"startTime":3071.01,"endTime":3074.34,"body":"the jackpot if we win."},{"startTime":3074.34,"endTime":3077.43,"body":"So, you know, people often"},{"startTime":3074.34,"endTime":3077.43,"body":"choose dates or memorable things,"},{"startTime":3077.43,"endTime":3078.96,"body":"anniversaries, birthdays,"},{"startTime":3078.96,"endTime":3082.2,"body":"so days that can be days"},{"startTime":3078.96,"endTime":3082.2,"body":"of the month one to 31"},{"startTime":3082.2,"endTime":3085.2,"body":"may be coming up more"},{"startTime":3082.2,"endTime":3085.2,"body":"often in people's choices,"},{"startTime":3085.2,"endTime":3087.81,"body":"not as draws in the"},{"startTime":3085.2,"endTime":3087.81,"body":"lottery, but what people do,"},{"startTime":3087.81,"endTime":3091.32,"body":"our sort of human way of"},{"startTime":3087.81,"endTime":3091.32,"body":"assigning significance to things."},{"startTime":3091.32,"endTime":3094.5,"body":"But don't think, well, I know that"},{"startTime":3094.5,"endTime":3095.55,"body":"one, two, three, four, five, six"},{"startTime":3095.55,"endTime":3097.83,"body":"is just as likely as"},{"startTime":3095.55,"endTime":3097.83,"body":"anything else, or unlikely,"},{"startTime":3097.83,"endTime":3099.3,"body":"so I'm going to show that I know this"},{"startTime":3099.3,"endTime":3100.89,"body":"and I'm a logical human being,"},{"startTime":3100.89,"endTime":3103.53,"body":"by putting the numbers one,"},{"startTime":3100.89,"endTime":3103.53,"body":"two, three, four, five, six."},{"startTime":3103.53,"endTime":3107.31,"body":"Because 10,000 people do"},{"startTime":3103.53,"endTime":3107.31,"body":"that every week. (laughs)"},{"startTime":3107.31,"endTime":3110.43,"body":"So if you do that, then"},{"startTime":3107.31,"endTime":3110.43,"body":"great; you are rational,"},{"startTime":3110.43,"endTime":3114.9,"body":"except you'll be sharing that"},{"startTime":3110.43,"endTime":3114.9,"body":"happiness of being logical"},{"startTime":3114.9,"endTime":3116.49,"body":"with 10,000 other people,"},{"startTime":3116.49,"endTime":3119.43,"body":"and you really do not want to"},{"startTime":3116.49,"endTime":3119.43,"body":"have to share a million pounds"},{"startTime":3119.43,"endTime":3122.13,"body":"or whatever the jackpot is, 10,000 ways."},{"startTime":3122.13,"endTime":3122.963,"body":"So don't do that."},{"startTime":3122.963,"endTime":3124.89,"body":"Final tip, final tip."},{"startTime":3124.89,"endTime":3126.29,"body":"This may be hard to achieve."},{"startTime":3127.26,"endTime":3129.99,"body":"Be Voltaire. (laughs)"},{"startTime":3127.26,"endTime":3129.99,"body":"(students laugh)"},{"startTime":3129.99,"endTime":3134.99,"body":"So Voltaire and his chum,"},{"startTime":3129.99,"endTime":3134.99,"body":"Charles Marie de La Condamine,"},{"startTime":3135.09,"endTime":3138.96,"body":"spotted a little loophole"},{"startTime":3135.09,"endTime":3138.96,"body":"in the French lottery."},{"startTime":3138.96,"endTime":3140.16,"body":"So the French government at that time,"},{"startTime":3140.16,"endTime":3142.679,"body":"they were selling government"},{"startTime":3140.16,"endTime":3142.679,"body":"bonds to raise money,"},{"startTime":3142.679,"endTime":3145.08,"body":"but the interest rate"},{"startTime":3142.679,"endTime":3145.08,"body":"wasn't very attractive."},{"startTime":3145.08,"endTime":3146.85,"body":"So to sweeten the deal,"},{"startTime":3146.85,"endTime":3148.8,"body":"they had a little lottery attached to it."},{"startTime":3148.8,"endTime":3151.62,"body":"And you could, if you"},{"startTime":3148.8,"endTime":3151.62,"body":"had one of these bonds,"},{"startTime":3151.62,"endTime":3154.59,"body":"you could, for a little bit extra,"},{"startTime":3154.59,"endTime":3158.64,"body":"so for actually 1,000th of"},{"startTime":3154.59,"endTime":3158.64,"body":"the price of the bond you had,"},{"startTime":3158.64,"endTime":3161.85,"body":"that would entitle you"},{"startTime":3158.64,"endTime":3161.85,"body":"to buy a lottery ticket."},{"startTime":3161.85,"endTime":3165.03,"body":"So if you had a 10 pound"},{"startTime":3161.85,"endTime":3165.03,"body":"bond, for one extra penny,"},{"startTime":3165.03,"endTime":3167.28,"body":"you could have a lottery ticket."},{"startTime":3167.28,"endTime":3169.29,"body":"The flaw in the plan was"},{"startTime":3169.29,"endTime":3172.41,"body":"that the prizes you"},{"startTime":3169.29,"endTime":3172.41,"body":"would win in the lottery"},{"startTime":3172.41,"endTime":3174.99,"body":"did not depend on what"},{"startTime":3172.41,"endTime":3174.99,"body":"you paid for your ticket."},{"startTime":3174.99,"endTime":3179.37,"body":"So Voltaire and La"},{"startTime":3174.99,"endTime":3179.37,"body":"Condamine started buying up"},{"startTime":3179.37,"endTime":3183.69,"body":"thousands of tiny, tiny, cheap,"},{"startTime":3179.37,"endTime":3183.69,"body":"there was no minimum price,"},{"startTime":3183.69,"endTime":3187.56,"body":"tiny little bonds, and each"},{"startTime":3183.69,"endTime":3187.56,"body":"one of those tiny, tiny bonds"},{"startTime":3187.56,"endTime":3189.18,"body":"gave them an equal chance"},{"startTime":3189.18,"endTime":3191.49,"body":"with everybody else in this lottery."},{"startTime":3191.49,"endTime":3195.69,"body":"And they did this successfully"},{"startTime":3191.49,"endTime":3195.69,"body":"over several months,"},{"startTime":3195.69,"endTime":3197.61,"body":"and they won a lot of money."},{"startTime":3197.61,"endTime":3199.02,"body":"They made huge profits,"},{"startTime":3199.02,"endTime":3202.05,"body":"about equivalent of 6 million"},{"startTime":3199.02,"endTime":3202.05,"body":"pounds-ish in today's money."},{"startTime":3202.05,"endTime":3204.69,"body":"And the government got"},{"startTime":3202.05,"endTime":3204.69,"body":"a bit cross, (laughs)"},{"startTime":3204.69,"endTime":3207.57,"body":"and they actually tried to prosecute them,"},{"startTime":3207.57,"endTime":3209.107,"body":"and took 'em to court and"},{"startTime":3207.57,"endTime":3209.107,"body":"said, \"You can't do this."},{"startTime":3209.107,"endTime":3211.86,"body":"\"This isn't what we meant.\""},{"startTime":3211.86,"endTime":3213.81,"body":"But the court ruled in their favor,"},{"startTime":3213.81,"endTime":3215.16,"body":"in Voltaire and La Condamine's favor."},{"startTime":3215.16,"endTime":3217.417,"body":"They said, \"Listen, it's not illegal"},{"startTime":3217.417,"endTime":3220.488,"body":"\"to exploit government's"},{"startTime":3217.417,"endTime":3220.488,"body":"stupidity,\" (laughs) right?"},{"startTime":3220.488,"endTime":3221.97,"body":"(students laugh)"},{"startTime":3220.488,"endTime":3221.97,"body":"Your own fault."},{"startTime":3221.97,"endTime":3224.85,"body":"So they got to keep their winnings,"},{"startTime":3224.85,"endTime":3226.41,"body":"and Voltaire became a rich man,"},{"startTime":3226.41,"endTime":3227.82,"body":"and was perhaps able to devote more time"},{"startTime":3227.82,"endTime":3230.85,"body":"to writing and philosophy, so it's great."},{"startTime":3230.85,"endTime":3235.38,"body":"But so yeah, be Voltaire, is"},{"startTime":3230.85,"endTime":3235.38,"body":"a good tip in life in general."},{"startTime":3235.38,"endTime":3238.89,"body":"Okay, so we've reached the"},{"startTime":3235.38,"endTime":3238.89,"body":"end of my little guided tour"},{"startTime":3238.89,"endTime":3241.23,"body":"of the lottery, and chance and fate."},{"startTime":3241.23,"endTime":3243.69,"body":"I'll just mention if you"},{"startTime":3241.23,"endTime":3243.69,"body":"would like to come back here"},{"startTime":3243.69,"endTime":3246.54,"body":"on March the seventh, and if you do,"},{"startTime":3246.54,"endTime":3248.61,"body":"you will hear about the mathematical life"},{"startTime":3248.61,"endTime":3249.99,"body":"of Sir Christopher Wren."},{"startTime":3249.99,"endTime":3251.85,"body":"But I will stop there;"},{"startTime":3249.99,"endTime":3251.85,"body":"thank you very much."},{"startTime":3251.85,"endTime":3254.767,"body":"(students applaud)"},{"startTime":3264.794,"endTime":3266.4,"body":"- Lovely; thank you very much, Sarah."},{"startTime":3266.4,"endTime":3268.14,"body":"That was excellent, as always."},{"startTime":3268.14,"endTime":3270.24,"body":"I shall now have to accuse anyone"},{"startTime":3270.24,"endTime":3271.62,"body":"who's buying a Lotto ticket in front of me"},{"startTime":3271.62,"endTime":3274.8,"body":"to stop and be rational,"},{"startTime":3271.62,"endTime":3274.8,"body":"but not too rational,"},{"startTime":3274.8,"endTime":3276.54,"body":"about the whole process."},{"startTime":3276.54,"endTime":3279.78,"body":"I don't know if anybody here"},{"startTime":3276.54,"endTime":3279.78,"body":"wants to ask a question."},{"startTime":3279.78,"endTime":3280.95,"body":"The only online one we've got"},{"startTime":3280.95,"endTime":3282.824,"body":"is about card counters in Las Vegas."},{"startTime":3282.824,"endTime":3283.657,"body":"- Uh-huh."},{"startTime":3283.657,"endTime":3285.93,"body":"- And are they"},{"startTime":3283.657,"endTime":3285.93,"body":"mathematicians extraordinaire"},{"startTime":3285.93,"endTime":3289.11,"body":"and do they know the"},{"startTime":3285.93,"endTime":3289.11,"body":"probabilities just like that?"},{"startTime":3289.11,"endTime":3293.04,"body":"- So what is not happening,"},{"startTime":3289.11,"endTime":3293.04,"body":"the card counters,"},{"startTime":3293.04,"endTime":3296.82,"body":"they are not having sort of"},{"startTime":3293.04,"endTime":3296.82,"body":"prodigious feats of memory"},{"startTime":3296.82,"endTime":3299.37,"body":"and actually memorizing every single card,"},{"startTime":3299.37,"endTime":3301.29,"body":"and adding up numbers."},{"startTime":3301.29,"endTime":3305.04,"body":"What most of the systems"},{"startTime":3301.29,"endTime":3305.04,"body":"that are there assign a value"},{"startTime":3305.04,"endTime":3308.04,"body":"of something like plus-one"},{"startTime":3305.04,"endTime":3308.04,"body":"to every low card,"},{"startTime":3308.04,"endTime":3309.33,"body":"say, between two and six,"},{"startTime":3309.33,"endTime":3311.16,"body":"and then zero to seven, eight and nine,"},{"startTime":3311.16,"endTime":3313.527,"body":"and then minus-one to 10 and upwards."},{"startTime":3313.527,"endTime":3315.84,"body":"And so you've got a"},{"startTime":3313.527,"endTime":3315.84,"body":"running tally in your head,"},{"startTime":3315.84,"endTime":3317.49,"body":"which you're only adding one at most"},{"startTime":3317.49,"endTime":3319.41,"body":"or taking away one each time."},{"startTime":3319.41,"endTime":3321.84,"body":"And then there are triggers of like,"},{"startTime":3321.84,"endTime":3323.64,"body":"if it gets above this amount,"},{"startTime":3323.64,"endTime":3325.47,"body":"that means lots of small"},{"startTime":3323.64,"endTime":3325.47,"body":"cards have been played,"},{"startTime":3325.47,"endTime":3326.58,"body":"so it's favorable."},{"startTime":3326.58,"endTime":3328.32,"body":"So there's sort of trigger"},{"startTime":3326.58,"endTime":3328.32,"body":"points where you then,"},{"startTime":3328.32,"endTime":3329.865,"body":"okay, I'm going to make a bigger bet."},{"startTime":3329.865,"endTime":3331.2,"body":"They have to learn these things."},{"startTime":3331.2,"endTime":3332.033,"body":"They are aware,"},{"startTime":3332.033,"endTime":3334.14,"body":"or at least the people"},{"startTime":3332.033,"endTime":3334.14,"body":"who design the systems,"},{"startTime":3334.14,"endTime":3336.18,"body":"are aware of the exact probabilities."},{"startTime":3336.18,"endTime":3337.86,"body":"You don't have to necessarily know"},{"startTime":3337.86,"endTime":3340.41,"body":"those probabilities"},{"startTime":3337.86,"endTime":3340.41,"body":"yourself to use the system,"},{"startTime":3340.41,"endTime":3343.35,"body":"but what you do have to"},{"startTime":3340.41,"endTime":3343.35,"body":"be, and this is less about"},{"startTime":3343.35,"endTime":3345.69,"body":"actually once you're playing,"},{"startTime":3343.35,"endTime":3345.69,"body":"when you're playing it,"},{"startTime":3345.69,"endTime":3349.23,"body":"it's less about mathematics"},{"startTime":3345.69,"endTime":3349.23,"body":"and more about nerve, I think,"},{"startTime":3349.23,"endTime":3351.75,"body":"and having a very good"},{"startTime":3349.23,"endTime":3351.75,"body":"memory even under pressure."},{"startTime":3351.75,"endTime":3353.7,"body":"'Cause you've got to keep"},{"startTime":3351.75,"endTime":3353.7,"body":"this tally in your head."},{"startTime":3353.7,"endTime":3355.83,"body":"If you're up to 17 and then you"},{"startTime":3353.7,"endTime":3355.83,"body":"get distracted by something,"},{"startTime":3355.83,"endTime":3356.76,"body":"you know, where was I?"},{"startTime":3356.76,"endTime":3358.05,"body":"Oh, no, it's all gone wrong."},{"startTime":3358.05,"endTime":3359.46,"body":"Right, so you don't want"},{"startTime":3358.05,"endTime":3359.46,"body":"to be in that position."},{"startTime":3359.46,"endTime":3362.13,"body":"'Cause as I've said, the"},{"startTime":3359.46,"endTime":3362.13,"body":"actual advantage you get"},{"startTime":3362.13,"endTime":3365.94,"body":"even playing perfectly is"},{"startTime":3362.13,"endTime":3365.94,"body":"very, very small and marginal."},{"startTime":3365.94,"endTime":3369.0,"body":"So yeah, it's to design the system,"},{"startTime":3369.0,"endTime":3371.28,"body":"you do need to know the"},{"startTime":3369.0,"endTime":3371.28,"body":"probabilities and the mathematics."},{"startTime":3371.28,"endTime":3374.34,"body":"And so there was a group"},{"startTime":3371.28,"endTime":3374.34,"body":"of PhD students from MIT"},{"startTime":3374.34,"endTime":3376.44,"body":"a few years ago who,"},{"startTime":3374.34,"endTime":3376.44,"body":"they'd manage to do this"},{"startTime":3376.44,"endTime":3377.67,"body":"and made some money on it."},{"startTime":3377.67,"endTime":3379.8,"body":"But really, (laughs) you know,"},{"startTime":3379.8,"endTime":3383.52,"body":"it is really, really difficult to do that,"},{"startTime":3383.52,"endTime":3385.68,"body":"and stay cool and not get too excitable"},{"startTime":3385.68,"endTime":3387.487,"body":"so that their people at the casino go,"},{"startTime":3387.487,"endTime":3390.234,"body":"\"Hmm, maybe that person,"},{"startTime":3387.487,"endTime":3390.234,"body":"we don't want them"},{"startTime":3390.234,"endTime":3392.013,"body":"\"playing in the casino,\" yeah."},{"startTime":3392.931,"endTime":3394.106,"body":"- [Student] You mentioned earlier"},{"startTime":3394.106,"endTime":3398.058,"body":"that if you're tossing a coin, say,"},{"startTime":3398.058,"endTime":3400.05,"body":"and you get a long sequence of heads-"},{"startTime":3400.05,"endTime":3401.04,"body":"- Yeah."},{"startTime":3401.04,"endTime":3403.103,"body":"- [Student] And some"},{"startTime":3401.04,"endTime":3403.103,"body":"people think erroneously"},{"startTime":3403.103,"endTime":3406.754,"body":"that a tail must be due,"},{"startTime":3403.103,"endTime":3406.754,"body":"if you're a statistician,"},{"startTime":3406.754,"endTime":3411.317,"body":"you begin to query the-"},{"startTime":3406.754,"endTime":3411.317,"body":"- Yes."},{"startTime":3411.317,"endTime":3414.75,"body":"- [Student] Whether the"},{"startTime":3411.317,"endTime":3414.75,"body":"coin is truly balanced."},{"startTime":3414.75,"endTime":3417.57,"body":"When did that start"},{"startTime":3414.75,"endTime":3417.57,"body":"coming into mathematics,"},{"startTime":3417.57,"endTime":3420.87,"body":"the statistics rather than"},{"startTime":3417.57,"endTime":3420.87,"body":"the probability theory?"},{"startTime":3420.87,"endTime":3422.579,"body":"- Yeah, so I mean, that's a good point."},{"startTime":3422.579,"endTime":3425.982,"body":"If you did toss a coin"},{"startTime":3422.579,"endTime":3425.982,"body":"and it came up heads"},{"startTime":3425.982,"endTime":3430.327,"body":"92 times in a row, as actually"},{"startTime":3425.982,"endTime":3430.327,"body":"that happens in the play,"},{"startTime":3430.327,"endTime":3432.99,"body":"\"Rosencrantz and Guildenstern"},{"startTime":3430.327,"endTime":3432.99,"body":"are Dead\" by Tom Stoppard,"},{"startTime":3432.99,"endTime":3434.82,"body":"that sort of meditation of probability."},{"startTime":3434.82,"endTime":3437.76,"body":"In that case, they start to"},{"startTime":3434.82,"endTime":3437.76,"body":"suspect something's going on,"},{"startTime":3437.76,"endTime":3439.5,"body":"'cause it's heads 92 times in a row."},{"startTime":3439.5,"endTime":3441.51,"body":"But in their situation,"},{"startTime":3439.5,"endTime":3441.51,"body":"yeah, what's happening"},{"startTime":3441.51,"endTime":3443.1,"body":"is they aren't the"},{"startTime":3441.51,"endTime":3443.1,"body":"masters of their own fate,"},{"startTime":3443.1,"endTime":3444.06,"body":"because unbeknownst to them,"},{"startTime":3444.06,"endTime":3447.39,"body":"they're trapped as minor"},{"startTime":3444.06,"endTime":3447.39,"body":"characters in someone else's play."},{"startTime":3447.39,"endTime":3450.09,"body":"(laughs) But hopefully,"},{"startTime":3447.39,"endTime":3450.09,"body":"that's not our situation."},{"startTime":3450.09,"endTime":3453.15,"body":"If that did happen, as you say, a lot,"},{"startTime":3453.15,"endTime":3455.73,"body":"heads 25 times in a row or lots and lots"},{"startTime":3455.73,"endTime":3459.24,"body":"over a long period of time,"},{"startTime":3455.73,"endTime":3459.24,"body":"you would start to suspect"},{"startTime":3459.24,"endTime":3461.76,"body":"there's something up with this coin."},{"startTime":3461.76,"endTime":3466.05,"body":"Now, statistics as a"},{"startTime":3461.76,"endTime":3466.05,"body":"subject started to come in,"},{"startTime":3466.05,"endTime":3468.6,"body":"and those kind of analyses"},{"startTime":3466.05,"endTime":3468.6,"body":"over long periods of time,"},{"startTime":3468.6,"endTime":3470.52,"body":"and testing things and hypothesis testing,"},{"startTime":3470.52,"endTime":3472.92,"body":"that sort of thing,"},{"startTime":3470.52,"endTime":3472.92,"body":"statistics as a subject,"},{"startTime":3472.92,"endTime":3476.4,"body":"I mean, it got its name, I"},{"startTime":3472.92,"endTime":3476.4,"body":"guess, in the 19th Century,"},{"startTime":3476.4,"endTime":3478.29,"body":"it comes from the German word \"statistic,\""},{"startTime":3478.29,"endTime":3481.38,"body":"but that kind of means"},{"startTime":3478.29,"endTime":3481.38,"body":"numbers to do with the state."},{"startTime":3481.38,"endTime":3483.06,"body":"You know, so measuring things."},{"startTime":3483.06,"endTime":3484.47,"body":"Initially, it was just measuring things"},{"startTime":3484.47,"endTime":3487.08,"body":"like how many people have"},{"startTime":3484.47,"endTime":3487.08,"body":"we got in our population?"},{"startTime":3487.08,"endTime":3488.7,"body":"What's our production of wheat?"},{"startTime":3488.7,"endTime":3490.53,"body":"And then people started to think,"},{"startTime":3490.53,"endTime":3492.69,"body":"oh, we've got all these numbers,"},{"startTime":3492.69,"endTime":3495.69,"body":"we can actually perhaps spot some trends"},{"startTime":3495.69,"endTime":3496.56,"body":"and do some analysis."},{"startTime":3496.56,"endTime":3498.78,"body":"And so that statistics"},{"startTime":3496.56,"endTime":3498.78,"body":"comes from, originally,"},{"startTime":3498.78,"endTime":3502.65,"body":"it was just collation of"},{"startTime":3498.78,"endTime":3502.65,"body":"numbers to do with, you know,"},{"startTime":3502.65,"endTime":3504.48,"body":"yeah, a country, a state."},{"startTime":3504.48,"endTime":3506.73,"body":"But then in the 19th Century,"},{"startTime":3506.73,"endTime":3508.41,"body":"people like Florence Nightingale,"},{"startTime":3508.41,"endTime":3510.21,"body":"you start to get actual analysis of it."},{"startTime":3510.21,"endTime":3512.25,"body":"So that statistics as a thing"},{"startTime":3512.25,"endTime":3514.53,"body":"comes a lot later than probability,"},{"startTime":3514.53,"endTime":3516.723,"body":"like three, two, three centuries later."},{"startTime":3517.95,"endTime":3519.75,"body":"- [Student] Hi, there;"},{"startTime":3517.95,"endTime":3519.75,"body":"thanks for the talk."},{"startTime":3519.75,"endTime":3521.114,"body":"I was thinking about what you said"},{"startTime":3521.114,"endTime":3525.09,"body":"that we seem to understand ourselves"},{"startTime":3525.09,"endTime":3525.923,"body":"to be really poor, what"},{"startTime":3525.09,"endTime":3525.923,"body":"am I trying to say?"},{"startTime":3531.252,"endTime":3532.085,"body":"- We have a poor intuitive-"},{"startTime":3532.085,"endTime":3533.675,"body":"- [Student] A few things we understood"},{"startTime":3533.675,"endTime":3537.63,"body":"to understand probability poorly."},{"startTime":3537.63,"endTime":3538.463,"body":"- Yeah."},{"startTime":3538.463,"endTime":3539.46,"body":"- [Student] And we can't"},{"startTime":3538.463,"endTime":3539.46,"body":"rationalize around this,"},{"startTime":3539.46,"endTime":3541.361,"body":"and things like that."},{"startTime":3541.361,"endTime":3544.56,"body":"But I'm wondering whether, I"},{"startTime":3541.361,"endTime":3544.56,"body":"suspect it's not an accident"},{"startTime":3544.56,"endTime":3545.657,"body":"that we are like we are."},{"startTime":3545.657,"endTime":3548.4,"body":"'Cause we're talking about the probability"},{"startTime":3548.4,"endTime":3550.29,"body":"of things which relate to technology,"},{"startTime":3550.29,"endTime":3553.38,"body":"and we didn't evolve with"},{"startTime":3550.29,"endTime":3553.38,"body":"any kind of technology"},{"startTime":3553.38,"endTime":3554.95,"body":"before games or before"},{"startTime":3553.38,"endTime":3554.95,"body":"money or anything like that."},{"startTime":3554.95,"endTime":3555.783,"body":"- Yeah, yeah."},{"startTime":3555.783,"endTime":3556.616,"body":"- [Student] And do you"},{"startTime":3555.783,"endTime":3556.616,"body":"think it's because of"},{"startTime":3556.616,"endTime":3561.614,"body":"the status of the natural"},{"startTime":3556.616,"endTime":3561.614,"body":"world that maybe we do expect"},{"startTime":3562.5,"endTime":3565.59,"body":"seeing a correlation in things"},{"startTime":3562.5,"endTime":3565.59,"body":"and a lot less randomness?"},{"startTime":3565.59,"endTime":3567.54,"body":"So we are how we are"},{"startTime":3565.59,"endTime":3567.54,"body":"actually supposed to be."},{"startTime":3567.54,"endTime":3569.491,"body":"It's not our fault that we're like this."},{"startTime":3569.491,"endTime":3570.386,"body":"- Oh, yeah, okay. (laughs)"},{"startTime":3570.386,"endTime":3571.219,"body":"- [Student] Do you see what I mean?"},{"startTime":3571.219,"endTime":3572.07,"body":"- Yeah, I see what you mean."},{"startTime":3572.07,"endTime":3574.83,"body":"So it's this sort of actually, yes,"},{"startTime":3574.83,"endTime":3579.83,"body":"our ability to assess risk and things,"},{"startTime":3580.2,"endTime":3582.42,"body":"randomness is perhaps less prevalent"},{"startTime":3582.42,"endTime":3583.45,"body":"in the natural world just because-"},{"startTime":3583.45,"endTime":3585.81,"body":"- [Student] Maybe because"},{"startTime":3583.45,"endTime":3585.81,"body":"we're competitive-"},{"startTime":3585.81,"endTime":3588.627,"body":"- I mean, yeah, I think"},{"startTime":3585.81,"endTime":3588.627,"body":"there's something to that."},{"startTime":3588.627,"endTime":3591.54,"body":"And I suppose my feeling is, you know,"},{"startTime":3591.54,"endTime":3596.54,"body":"the penalty of underestimating"},{"startTime":3591.54,"endTime":3596.54,"body":"risk if you're just,"},{"startTime":3597.63,"endTime":3599.374,"body":"you know, a hunter-gatherer,"},{"startTime":3599.374,"endTime":3603.72,"body":"is different from the penalty"},{"startTime":3599.374,"endTime":3603.72,"body":"of overestimating risk."},{"startTime":3603.72,"endTime":3605.527,"body":"If you underestimate risk, go,"},{"startTime":3605.527,"endTime":3609.54,"body":"\"Oh, it's probably not a"},{"startTime":3605.527,"endTime":3609.54,"body":"tiger, (laughs) I'm fine,\""},{"startTime":3609.54,"endTime":3613.41,"body":"then that, you know, getting"},{"startTime":3609.54,"endTime":3613.41,"body":"that wrong is unfortunate."},{"startTime":3613.41,"endTime":3615.57,"body":"Or you could say, \"It might"},{"startTime":3613.41,"endTime":3615.57,"body":"be a tiger\" for everything,"},{"startTime":3615.57,"endTime":3616.74,"body":"okay, you're a bit panicky,"},{"startTime":3616.74,"endTime":3618.51,"body":"but you're less likely"},{"startTime":3616.74,"endTime":3618.51,"body":"to be eaten by a tiger."},{"startTime":3618.51,"endTime":3623.22,"body":"So I think we're perhaps"},{"startTime":3618.51,"endTime":3623.22,"body":"something about the way we've,"},{"startTime":3623.22,"endTime":3627.54,"body":"as we are animals after all,"},{"startTime":3623.22,"endTime":3627.54,"body":"there's less of a penalty"},{"startTime":3627.54,"endTime":3629.25,"body":"for overestimating risk"},{"startTime":3629.25,"endTime":3631.632,"body":"than there is for underestimating it."},{"startTime":3631.632,"endTime":3634.62,"body":"I'm not a biologist,"},{"startTime":3631.632,"endTime":3634.62,"body":"but that's my suspicion,"},{"startTime":3634.62,"endTime":3635.453,"body":"for sure, yeah."},{"startTime":3636.36,"endTime":3637.5,"body":"- Thank you."},{"startTime":3637.5,"endTime":3639.18,"body":"I would like to ask you all to join me"},{"startTime":3639.18,"endTime":3641.276,"body":"in thanking Sarah very"},{"startTime":3639.18,"endTime":3641.276,"body":"much for being here."},{"startTime":3641.276,"endTime":3644.276,"body":"(students applaud)"}]}