In Our Time - Probability

Episode Date: May 29, 2008

Melvyn Bragg and guests discuss the strange mathematics of probability where heads or tails is a simple question with a far from simple answer. Gambling may be as old as the hills but probability as a mathematical discipline is a relative youngster. Probability is the field of maths relating to random events and, although commonplace now, the idea that you can pluck a piece of maths from the tumbling of dice, the shuffling of cards or the odds in the local lottery is a relatively recent and powerful one. It may start with the toss of a coin but probability reaches into every area of the modern world, from the analysis of society to the decay of an atom. With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews; Ian Stewart, Professor of Mathematics at the University of Warwick

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio four. I hope you enjoy the programme. Hello. Heads or tails? It's a simple question with a far from simple answer, one that takes us into the strange and complex world of probability.

Probability is the field of maths relating to random events sand, although commonplace now, the idea that you can pluck a piece of maths from the tumbling of dice, the shuffling of cards, or the odds in the local lottery, is a relatively recent and powerful one. It may start with the toss of a coin, but probability reaches into every area of the modern world, from the analysis of society to the decay of an atom, to predicting the weather. With me to discuss probability are Ian Stewart, Professor of Mathematics at Warwick University, Colmeroni Dougal, lecturer in pure mathematics at the University of St. Andrews, and Marcus Usoitoi, Professor of Mathematics at Wardham College, University of Oxford.

Marcus Uso-Toy, dice cards, gambling. There are ancient pursuits, but who was the first person in the scheme of the setting off of this area of interest who got to grips with it? I think it's quite striking how late it is. It's actually an Italian mathematician called Cardano in the 16th century. I mean, as you say, we've been playing dice games for 5,000 years. There's a game which you can see in the British Museum which uses tetrahedral-shaped dice. But the idea that you can actually apply mathematics

to the role of a dice and actually, you know, you think, well, it's a random thing. How can maths, such a certain topic, be used to actually predict what a dice is going to do? But what Cardano realized that, yes, you can actually mathematically this randomness. And what he did was to see, well, you know, a dice, what's the problem? He was a heavy gambler himself.

He was a heavy gambler, so he was interested in actually trying to work out What effect this? Whether he could use some sort of logic to understand what he should bet on and what he shouldn't bet on. So I think most people have a very clear idea that to get a six, there's a one in six chance. But that idea of giving a fraction, a sixth, to the outcome of the role of a dice, is quite a major step forward, which he made. Why is it a major step forward? Because it seems blindingly obvious.

It seems so blindingly obvious. But then you sort of push the problem a little bit further. What about throwing two dice? What's the probability of getting it? a seven with two dice. Well, now it's starting to get a little bit sort of unclear, so of what... And this is what Cardano realizes, what you've got to realise is, what are all the different possibilities of two dice? There are 36 different possibilities of the two dice.

There's six choices of the first, six choice for the second, six times six, 36 different ways that the dice can land. And then here I, what you need to do, the winning ways to get a seven, how many different ways are that you can get a seven? Well, there are six different combinations, one and a six, a two and a five, a three and a four, a four and a three, a five and a two, a six and a one. So what do you realise is that six out of the 36 possibilities will give me a seven. So then he realized, well, again, the probability, the chances of getting a seven with a roll of two dice is one in six again.

But what about the chance of getting a double six? That's only one out of the 36 possibilities who will give you that. So it's starting to build up a sort of mathematics of actually understanding the role of the dice and saying, well, okay, it's more likely you're going to get a seven than, say, a 12. Can I take one brief step back? Your opening remark that it was surprising that it arrived so late. Was this to do with the religious connotations, both all sorts of religions connotations,

because they used to play dice with bones, didn't they? And these were looking into the future, and it had a religious, magical element in it. Yes, but you would still think that perhaps, you know, somebody would get wise the fact you can make some, predictions about the way these knuckle bones. I mean, that's actually these knuckle bones are the first sort of dice because they landed on sort of four sides and where you get these tetrahedral dice in the game of ore in Babylon.

But I think there was, the start, the ancient Greeks, the mathematics of the ancient Greeks, was a lot about certainty and proof. And it wasn't believed that this could be applied. So I don't think the ancient Greeks really thought of actually applying mathematics to the idea of something random. It just seemed ridiculous. And then I think sort of Christian, the Christianity, which kicked in as well, they weren't particularly pro sort of analysing the Theravadis. It was sort of something determined by God, not by mathematics. So I think it is quite a, that's one of the reasons

it came in so late with Cardano. But actually Cardano wrote a 15-page paper about this, which never got published until about 100 years later. So actually, it's two other mathematicians that are generally credited with the great sort of breakthroughs on probability, which are a very famous one, Fermat's last theorem, and Pascal and this came out of a problem One day we're talking about this?

Well we're talking we're talking middle of the 17th century so 100 years after Cardano and they hadn't access to his work it's interesting the way they didn't

the way that knowledge isn't transmitted sometimes Yes exactly but again it was gambling which gave rise to the breakthroughs that they made on probability Parisian gambler called

The Chevalier de Merre was quite interested in what happens when if you're playing a game game and the game is interrupted halfway through. How should you divide the winnings? So here's a particular game. You're flipping a coin and Ferramar wins if you get a tail and Pascal wins if you get a head and they've got to get six heads or six tails to win. But what happens if they break the game where Pascal has five heads, Ferramat has three tails, there's a pot of

20 pounds or something, how should they divide the pot? How much should Ferramat get? He's quite a long way from winning, but Pascal's only one away. Well, if you think, what does Ferramat got to do? He's got to get three tails in a row. Okay, so three tails in a row. How many other ways could three coins land, heads or tails? Well, there are eight different possibilities. The first coin could end heads or tails, second heads or tails, third heads or tails. So that means that one out of the eight possibilities will give Ferramma a win, but the other seven out of eight will give Pascal a win. So Pascal and Ferramar realised that the way to divide the winnings is that Pascal should get seven-eighths of the pot

and Ferramar only one-eighth of the pot. So there was a way that they could use probability to actually predict how things would, it might have worked out to actually divide the pot. So, raging into mathematics through gambling. Over Rone Dugel, this coin still being flipped. What's the reasonable explanation? If I've had five heads in a row,

we all think back to Tom Stoppard at this moment if I have five heads in a row most people including myself expecting tails to turn up next but what is, is that a reasonable expectation? No it's not so what's happening here is a concept called independence of different random events

now this was first invented not so long after the Pascal letters sort of end of the 17th century by a guy called Des Moirev now he was a Frenchman who was a Protestant and the edict of Nant got revoked, so he got kicked out of France for being Protestant and came to London, decided to learn some mathematics

and wasn't able to get a job in British universities because he was French. He hadn't been able to get a job in France because he was Protestant. Bad luck seemed to follow him around. So he spent a lot of time sitting in coffee houses helping people sort out odds for gambling games. This was one of his ways of making an income.

People would pay him to calculate the odds. So he realised that with a very event, like this, there's no causal relationship between what the coin has done the time before and what the coin is going to do the next time. The coin has no way of remembering the previous event. One way to make that clearer in your mind is to imagine, suppose you tossed a coin five times and it came up heads and you then carefully put it into a box and didn't touch it for 20 years. 20 years later you're going to open this box and pull out the coin. Now it becomes obvious that the coin isn't going to be any more likely to wind up being tails than it would be for heads.

So these particular things have no way of influencing each other one way or another. So how does theory get into that kind of randomness in that case, for instance? Well, there's a variety of ways in which events can be related. So independence means there's no connection between them at all. The next concept which people came up with slightly later is conditional probability, and that's where one event happening makes another one slightly more likely or slightly less likely to happen. Now, curious enough, you can illustrate this, because I've read your stuff, you can illustrate this by reference to an American game show.

Sure. So before I launch into this explanation, I should say at least half of the audience are going to think I've got this wrong, but trust me, this is the correct answer. So there's a game show. So we're talking about conditional probability. Conditional probability, where events are now connected, they can make each other more likely to happen or less likely to happen. The simplest is straight causation.

As distinct from independence, where you can do nothing about a coin. And it does what he wants to do. Indeed. So now one thing's going to make something else more or less likely. So this game show, you walk in, there's three doors. Behind two of these doors is a goat. This is the Monty Hall problem. Yes.

Monty Hall. We'll give him his full due. I mean, he's entered into high mathematics, Monty Hall. He never expected that when he set off. Yes, although we should mention that the game has played on the show is not the same as the game has played in the mathematical problem. And that's maybe one reason for the confusion. But in this imaginary version of the Monti Hall show, There's three doors. Behind two of them are goats, and we will assume for the sake of argument that you're not interested in getting a goat.

And behind one of them is a car. We'll assume that you're quite interested in getting a car. Now you get to pick a door. Free choice, any door you like. Once you have picked the door, and this bit's particularly important, the host will open one of the other two doors, and he's guaranteed to show you a goat. he knows where the car is and you are absolutely promised that he will not show you the car, even if it is behind one of the other two doors. The question that's then raised is, would it be better for you to stick with your original choice

or to switch to the door that's the one of the other two that's not yet been opened? Now, most people, when they first hear this problem, think now, I've got two doors, these are independent events, it's 50-50, it doesn't matter. What we're missing here is that additional information has come into play since you first made your choice. So initially, all three doors, these were independent choices, you had probability one-third for each of them. When he opens a door, there's a two-thirds probability that you didn't pick the right door in the first place.

When he opens one of the doors, the whole of that two-thirds probability now has to sit behind the door that he didn't open. So you're always better to switch rather than a sticking. as you say. Many people are scratching their heads here. So we'll move on swiftly to another head scratcher. In the early 18th century, a man called Jacob Bernouli, a Belgian, one of eight Belgian mathematical brothers,

wrote the art of conjecturing. He established the idea of the Law of Large Numbers. Over to you, in Stuart. Law of large numbers relates the kind of version of probability that mathematicians were thinking of, which is, say, a fair coin, 50% heads, 50% tails.

probability a half for each. Probability is between nought and one. Nought is impossible, one is certain. So probability of a half means half the time you expect a head, half the time you expect a tail. Now he related that idea to what happens when you toss the coin lots and lots and lots of times. I suppose someone gives you a coin and says, is this a fair coin? Are the probabilities 50% each way? or maybe this has been subtly biased so it comes up heads twice as often as tails and you can't tell by looking at it there's hidden weights inside or something

so Bernoulli's idea was the way to find that you keep tossing it lots and lots of times and intuitively if I've got a fair coin and I toss it a hundred times I expect about 50 heads and about 50 tails and if it was biased I might get 70 heads and 30 tails okay it sounds easy but now at what point do you start deciding that

there's something a bit strange about this coin? I mean, I once tossed a coin 17 times in a row and got heads every time. This beats Tom Stoppard, I think. And it genuinely happened. This is wildly unlikely, but it was in fact a fair coin. So Bernoulli did the sums, and he had a whole kind of

technique for what are called Bernoulli trials. You know, it sounds a bit ominous, but it's just do the same thing over and over again, repeat the experiment many, many times. So suppose you have a coin and you toss it four times. There are 16 different sequences of heads and tails that could come up. One of those is heads, heads, heads. Four of them have three heads and a tail.

Six of them have two heads and two tails. Four of them have one head and three tails and one has four tails. So there's a nice pattern of numbers. One, four, six, four, one out of 16. and basically he worked out how these patterns go. Now there's an interesting feature of those numbers. What's the probability, if I tossed a coin four times and it's a fair coin, what is the probability of two heads and two tails?

What's the probability they really do split exactly as you'd expect? Well, it's six out of 16. What's the probability that it's three of one and one of the other? Well, it could be three heads and a tail, that's four, or three tails in a head, that's another four. That's eight out of 16. It's actually bigger. So you're more likely to get a three to one split

than you are to get a two to two split, even though it's a fair coin. Can I move to briskly to his... The black and white pebbles in the big jar, the 3,000 black and white pebbles in the big jar, which seems to be one of the key elements in his proofs. He was very hot on that model. To think about probability,

it helps to have some sort of... physical model. It's probably a rather idealised way. So you had these pebbled in a jar. Let's say 2,000 and white and 1,000 black. And you put your hand in blind. You picked a pebble out. You put it down and then you put it back in. You shook it up again. You picked it another. Then what happened? What was he proving? Okay. What he was proving

there was that if you sample with replacement, as they say, if you keep putting the pebbles back in the jar, then the probability of getting a black pebble or a white pebble is the same at every step. And I think he introduced the pebbles in a jar so he could get away from the 50-50.

You can tailor your pebbles to give any bias you like. But we know, because I know from your notes, that he could pull out 100 white pebbles. Yes, he could pull out. Well, it could pull out your 17 heads. I mean, such an example for us all. He could put out a thousand white pebbles. He could, but it gets...

So what does that do with this stuff? Okay, the probability of pulling out these rather unusual numbers, which don't seem to reflect the proportions, are very, very small. You can do this, for example, the probability of my 17 heads is a half times, a half times, a half times, a half, 17 times.

Okay, that's, I'm not going to do the sums, but that's getting pretty small. It's about 1 in 100,000 or so. So this is actually pretty unlikely. But you can quantify this in terms of your original numbers of balls in the urn, the number of black, the number of white. So you get a handle, not just on what's the average

proportion that turns up, which should go according to the proportions in the urn, but what are the fluctuations around the average? Can we stick to gambling as a way into this, Marcus? It started that with Godano. And we've talked about what became the law of large numbers. Ian has been talking about then. Most people know about a roulette wheel. How does it apply there?

Yeah, this is a very good example, because the roulette wheel, there are 37 numbers on a roulette wheel. but they pay out in a casino as if there are 36. So if you put one pound down on a particular number coming up, you get the £1 back and £35. So the point is that this is biased very slightly in favour of the casino. The zero is their number. The zero, yeah, is essentially their number.

I mean, you can bet on zero, but the point is you're betting the reward you're getting. It would be an evens game where you'd both come out equal if there were 36 numbers and you were paying out at, you know, one, 36 pounds for putting back one on a particular number. Now, the law of large numbers applies here, because you have the chance of actually winning a big amount by putting your money on the right number.

But in the long term, the casino will have a slight edge. So I think for each pound that's put down, that edge will give them, they'll earn about two pence. But over the long term, that adds up to a lot of money. So they don't mind paying out to, you know, one big winner who gets the chance to win a large amount, because over the long term, the bias will switch more and more to them, so that they're 2P per turn.

You're paying every time basically you play the game to have a chance, 2P to the chance of winning, sort of something big. So the large number is actually essential to a casino because it's the reason that they actually come out on top in the long term. And just to reassure listeners that this isn't a, this isn't a program entirely devoted to gambling and how to improve your own. This is the way that this massive new development mathematics

was led into the mainstream through analysis of various methods of gambling. Because here we have something called a gambler's fallacy as well, don't we? Oh, yeah, well, this is a great one. I mean, because actually, you know, I put my money down on it coming up as an even number. I put one pound down, and I lose. So next time round, I put down two pounds down.

So next if I get, I've already lost a pound, but if I win this time, I'll get two pounds back. So I'll be actually one pound up. If I lose again, which would be rather bad, but I lose again, I put four pounds down on evens. And each time you double your bet up. And eventually, you know, it can't be odd forever. So eventually it'll be even. And when it's even, because you've doubled your bet up each time, if you add up the money that you spend and the money you'll get back, you'll actually be one pound up. So, you know, in this way, this is a sure-far way of winning money because you just keep on doubling your money up and you'll eventually an even number will come up. You'll get a head, you'll get a pound back. This is the reason casinos have a maximum bet because at some point you cannot double your bet and you cannot go above the maximum bet on the table. So they're relying on the fact that at some point you will not have enough money to put down. You can't win and therefore the casino will also win. So that's quite important in actually making sure that the casino can't, you know, the math says you should be able to win all the time. Cool, but we want to come in, but there's something I want to ask you anyway. So you have to say, well. So there's a related gambling problem which was invented by one of the Bernoulli's who you mentioned earlier, which is known as the St. Petersburg paradox. So here again, it's a gambling game.

And the rules of the game are a coin is going to be tossed. If it comes up heads on the very first toss, you win a pound. If it comes up heads on the second, you win two pounds. If it comes up heads on the third, you win four pounds. If it comes up heads on the fourth you win eight pounds, it keeps doubling. The question is, what is a fair price to pay to be allowed to play this game? And this flummox probability theorist for a very long time, because if you do the maths, your expected possible gain is infinite, because this could keep going on forever.

There's no upper limit to the amount you could win here, which would suggest you ought to be willing to pay almost any quantity of money to play the game. But of course, in practice, you wouldn't be willing to pay any sum. You'd set a very definite upper limit, because there's a 50-50 chance you only get one pound back. So the way we understand this now is that, of course, there is no such thing as an infinite pool of money. It sort of doesn't make sense to talk about games like this,

particularly in terms of how would you feel about playing this game. And again, people are edging into this. That's one of the fascinating things about this. The way people edged into it, because I knew so little about it, it didn't seem to matter, it didn't seem possible it was random. And the birthday, how many people have to be in it? I feel like, I just sort of, I'm just,

keep saying to myself, look, this leads to quantum mechanics. I keep going. How many people need to be in a room to have the same birthday? And why does it matter? 23 is the answer. So I can briefly indicate how you work that out. We all get it completely wrong when we first think about this. The easiest way to do the sum is to turn it round and say how many people need to be in the room for none of them having no birthday to have the same birthday to have probability less than a half.

And one way you can think of that is the first person comes into the room. They've got their birthday. The next person comes into the room. Discounting leap year because they make things complicated. There's a 364 and 365 chance that they don't have the same birthday. The next one comes in. Well, now we're down to 363 because two days have been used up. And when you start doing that sum where the top part of the fraction goes down by one each time,

you find that it drops below a half much, much quicker than you'd expect. And actually at 23, it comes out at 6. slightly less than 50. I have got to allow Marcus to show. Yes, indeed. He's been waiting all the morning for this flourish on the birthday. 23 people. Right. There's a lovely football interpretation, which I knew Melvin

would love at this point. Which is, how many matches, premiership matches, do you expect there to be where two people on the pitch have the same birthday? Because you've got 11 players on each side, plus the referee, that makes 23 people on the pitch. So actually you expect over half the matches over the weekend to have

people with two people with the same birthday on the pitch at the same time. And Ian was telling me earlier that actually somebody did an experiment one weekend, and it did work out. I mean, I think the point about this is it shows how counterintuitive probability can be. And that's really the amazing thing. Probability tricks you every time. You would never believe it was as small as 23 people in a room for it to be more likely that two have the same birthday.

Yeah, I mean, I think that one of the things it says is that, coincidences are much more likely than you think if you don't specify in advance what coincidence you're thinking about and if you do the same calculation and ask a slightly different question how many people should be in the room for at least one of them to have the same birthday as me not the same birthday as somebody else in the room but a specific person then the number goes about massively it's much much bigger

it's much bigger than you'd expect it's about 250 so um our intuition about these things is wrong both times. It kind of overestimates one, underestimates the other. And in fact, I think our intuition about... You'll have the same birthday as somebody, but you can't claim it's going to the same birthday as you.

That's right. You know, somebody walks in and, you know, if we specify the birthday before we start doing that calculation, all the numbers are different, the sums are different, results wildly different. And now the fun stops in this programme because we start talking about the serious theory centering. Let's take the Kolmogorov, the Russian, who brought in axioms,

away you go, and briefly on the axioms, and we'll move forward from there. The sort of probabilities we've been talking about... This is a turn of the 19th into the 20th century. The whole thing cranks up a gear mathematically at this point, because there's a second area where probabilities can be applied, where everything we've talked about fails completely. Everything we've talked about has been some sort of finite number. of possibilities, six possibilities for rolling dice, and we count how many do what we want.

And it's all to do with this counting system, and you form some fraction of the number of ways what you want can happen out of the total number of ways. But suppose you're throwing a dart at a dart at a dartboard. Let's keep this very, very serious. You're throwing a data to dart at a dart, what's the probability you'll hit the bullseye? Well, mathematically there are infinitely many points inside the bullseye. And there are infinitely many points. points on the whole dartboard. And you can't just say infinity divided by infinity. So the

intuitive thing is that, well, it's, the ball size is pretty small that art boards a lot, is something to do with the areas of the bits and pieces of the dark board, let's say. But now we're talking about probabilities when there are infinitely many possibilities, not just doing

the same thing infinitely many times, but actually the outcomes are infinitely many. And mathematicians got themselves into a real tangle with this because all of the counting methods fail and there are some notorious problems where

you can calculate the answer in several plausible ways and you get different results. Yeah, we've got to get to this Russian and his axioms. Comagor, I said, forget all that. Forget all that. Let's go for the jugular. I will write down not what probability is in these

circumstances but what properties it ought to have. I'll separate the problem into two pieces. What do you want this thing to do and how can we build one? It's a bit like what's a spade. The useful answer, a spade is an implement for digging or garden.

This is the Cole McGorough answer. It's what you want to do with it and how it works. The less useful answer is it's a piece of metal attached to a length of of wood with a handle on the top. Why does this take us forward, Marcus Issaid, are these axioms? Well, it gives you the ability to actually

talk about probability in a sort of axiomatic way, which is what mathematics somehow has always been built on. The ancient Greeks did geometry by writing down properties of how lines join points and things like this. And from that, they could then develop a systematic, logical mathematics which, you know, couldn't be questioned. So you're taking a statement, you're making a statement and which you could, from which you could, you were, you were inventing a statement, as it were, from which you could logically prove various matters. Exactly. And it was very necessary. because probability is such a slippery subject

that you could often find yourself just coming up with contradictions and that's somehow an anathema to mathematicians and so what Colmogorov did was to produce an axiomatic framework which enabled us to actually move probability on in a mathematically consistent and logical way. Colbert, can we take this with reference

to the idea of the average man? This is Comorov. So that is an idea. There is an average man, is the axiom. Now then, how does that help probability. Okay, so we're jumping back in time slightly here, actually, to sort of early 19th century and a Belgian guy called Cotillet. Now, we were talking earlier about the law of large numbers and the way in which things tend to wind up coming out roughly around the average if you look at enough of

them. This had already been applied in cases where they were doing astronomical observations to try and correct errors in astronomical data. You would know that the correct answer was somewhere in the middle. What this Belgian Kucle did was he started applying this kind of theory to the human population and saying what is the average age at which people marry say and finding

little curves for that? What is the propensity of different groups of the population to commit crimes and finding curves for that? There was quite an outcry after his work that he seemed to be questioning the notion of free will that he was saying a certain number of people will commit crimes

in London on a Saturday night and what about these people's relationship with their morality and God? So it was the first time that probability was being used to make predictions about human behaviour and hence was entering the real world. So the application of probability markets just are, is in statistics? Yes, so at a certain extent. I mean, for example, this information was used to start the insurance industry. I mean, you want to work out, you need to know what, the probability that somebody is going to have.

a car accident or is going to die at a very early age and therefore you need to set your premium to be able to cover that you know how many people are likely to do something on the edges of the statistical data that I'll have to pay so how much do we need to charge the whole population so so so you start to see in 19th century 20th century probability and statistics really feeding in to to real world problems one thing I would say though is that this question of axiomatization is the sort of thing that we pure mathematicians get really excited about. It's something that the people out there doing applied statistics care quite a lot less about. What they're interest in is how to collect data, how to deal with influences in data, how to deal with biases of various different

types, whereas we would quite happily not really think about the interpretation of probability at all. We would say that probability is anything satisfying, a bunch of axioms that we're quite happy with and run off into mathematical never-level land. So there's two main schools of how to interpret probability, which is still very much arguing with each other nowadays as to what probability means. Frequentism and Bayesianism are the two big schools of the 20th century, I would say. Ian Stewart, I want to turn our attention to physics, but start with an 18th century physicist, Piersci Simon Laplace. How did he envisited the universe and the role of probabilities our understanding it? He, as it were, rather eloquently set out the agenda, didn't he?

Laplace is faced with the problem of, on the one hand, according to classical Newtonian mechanics, the entire universe is deterministic. There are mathematical rules. It follows the rules. And that means, again, this question is free will, or appears to. But it sort of, you feel that if something's following rigid rules, then it's completely deterministic,

then it can't actually do anything terribly complicated. And so Laplace rather eloquently said that if some vast intellect could actually encompass all these rules and observe the exact state of the universe right now, then the future would be completely predictable. And on the other hand, we know that in practice it ain't like that. It just is not like this. You really can't even look out of the window and predict whether it's going to rain in the next 10 minutes or not in many cases. So we live in this wildly unpredictable universe that runs on totally predictable rules. And this was a bit of a challenge to everybody.

It wasn't seen as a challenge at the time. What it turns out is that even deterministic rules can lead to wildly irregular behaviour. Can I just follow that? Probability seems to be really pushing in quite heavily, doesn't it? Making itself felt Marcus has referred to the insurance industry, which makes it a very serious matter for all of us. But can you explain how probability enable us to predict at the end of the 1930-20th century, the behaviour of gases.

They're far too complex for us really to understand them, as I understand it, but probability enabled us to get a grip on them, and therefore to move forward with studying them. So this is down to two people, James Clark Maxwell, who I'm sure lots of people will have heard of, and a guy called Boltzmann, who quite possibly fewer people will have heard of, who is European-Austrian. So at the time, whether or not matter was atomic was still a matter for some debate.

Most people were beginning to think that it was, but not everyone agreed. Now, one of the problems with that is if you're looking to analyse a gas, then what it must be, by the atomic theory, is a whole bunch of different little atoms whizzing around. And by a whole bunch, I mean an awful lot. I think one milligram of gas is going to contain something like 100 trillion particles. So an awful lot. Each of those particles is behaving entirely predictably,

except that it bumps into other ones all the time. So if you were to study just two or three gas molecules, you'd be fine. But by the time you want to study a cubic litre of oxygen, you've got a problem. So what Maxwell and Boltzmann first realized was, whilst they couldn't talk about each individual particle, they could talk about the average properties of these particles. They worked out that on average they're all travelling at the same speed, although there's some variety between each of them.

On average, they're equally likely to be travelling in any one direction to any other. So using these probabilities, they were then able to describe the way that gas behaves, and in fact they went on to use this to describe the way that heat flow behaves. There was a big problem with the second law of thermodynamics, because all of Newton's laws are time reversible. You can put in minus time rather than plus time, and it all works. But we know that entropy increases. So Boltzman was finally able to do.

the second law of thermodynamics by proving that what's happening is that with high probability entropy increases. This is why temperature differences average out. This is why in general the universe becomes more and more chaotic and uniform as time goes on, although obviously there's lots of local variations to that. So it really was the use of probability to describe aspects of the real world that could not be described by a fully deterministic system because it was too complicated. Can you bring quantum physics into this in the early 20th century market? It says to I, and as far

as I'm concerned, you're on your own here. Richard, well you've got two, you've got allies, a couple of allies. But as I'm saying, it's challenged prevailing ideas about determinism and probability. But probability is now part of the function of the new... This is an

incredible discovery, really, that probability is at the heart of the way just one particle works. I mean, what Colver's been talking about if you take a lot of particles, whether you knew where each of them were, you'd be able to describe the gas, but the best way to describe,

you know, you can't know the exact location of a trillion particles, so let's do a sort of average thing, and then we can work out what it should be doing. Quantum physics realized, well, if you look at just one electron, well, you should know what that's doing. It's like a little billiard ball.

That was the sort of conventional classical idea that was sort of whizzing around. But quantum physics realized, well, it doesn't work like that at all. Actually, if you want to know the location of a particular electron, it doesn't sort of make up its mind until you observe it. And then it can be sort of,

it might half the time be on sort of the left hand side of you and half the time on the right hand side of you. And each time you look, it makes a different decision. So probability suddenly became at the heart of just actually observing sort of even just one particle. What's interesting is that quantum physics actually is deterministic until you observe it. So we have something called the wave equation,

which is something which lives in the world of complex, numbers, when you look at this thing and observe it, it sort of has to collapse into real numbers, and this has the effect of then the wave function basically tells you, what's the probability that the electron will be at one place and not another? So it's almost built in to the physics that you have to look at it in a probabilistic way. And one example of this is radioactive material. So the way it decays is you can't actually make any predictions about when it's going to throw out a particle. It just behaves as a random process. And what's happened before, the decay of the uranium before, does not affect what's going to come after it, just in the way of the same as the flick of a

coin. Now, a lot of physicists, including Einstein, found this totally unacceptable. That's, you know, to have physics, something which we thought was so deterministic, predictable, based on a whole the theory of probability, has been deeply unsettling. But it's the best model we have so far. But Einstein believed that it wasn't going to be the final answer, and he believed that, you know, God does not play dice, is what he says. Do you want to take that on, Ian, sir? Yeah, it's fascinating, looking back, that Einstein used that particular metaphor,

because within that metaphor are the seeds of its own destruction. Because dice, as a metaphor for chance, dice are also a deterministic, classical, dynamical system. If I roll a cube, and if I knew exactly the speed, with which it hit the table and various other things, then in principle, maybe not in practice for various reasons, in principle I could actually calculate which side is going to come up. If I tossed a coin and knew just how fast I'd tossed it,

how fast it was spinning, I could say, that's going to be heads, and I'd be right. So our metaphor for probability has a sort of internal black box, inside the black box of probability is the classical dice. Now Einstein didn't think of it that way. he didn't mean it that way. But there are people who wonder whether actually maybe that's what's going on

in the quantum world. If I'm a radioactive atom sitting there waiting to decay according to some probability, how do I do that? How do I know when to decay? More subtly, how do I know how to get the proportions and the timings following the right probability law?

How is that probability law built into the physics? And what the physicists do is basically they've got so used to it being part and parcel of the physical world and their view of the physical world, they say, look, it's just like that. An electron is a bunch of probabilities, in a sense, for this purpose.

But it's possible that there is something inside the black box, it might be deterministic, it might be going back to what Laplace said, it could be deterministic, but still appear to be random because it's very irregular. Colber. Yeah, so picking up on what Ian was just saying,

there are various models posited which led us regain, determinism, but they all come with quite nasty costs. So possibly one that more people will have heard of is the many worlds theory, of which I guess the most famous exponent was Brent Everett. And what that posits is that at every moment with all of these random quantum events, different universes are being created for each one. So it's not in any way probabilistic. They all happen. At every split second, there's just trillions and trillions and trillions of completely different universes heading off in different directions. And the probability aspect,

comes in and that I don't know which one I'm going to wind up finding myself in. Of course, I'm in all of them, but I don't know which one it's going to feel like I find myself in. So that's one fix. The other fix is to posit hidden constants, hidden variables, which can sort of be made to work, but the problem

with that is you wind up violating causality. You wind up with action at a distance making things happen. So neither of those on their own feels much more emotionally satisfactory than letting probability come in. In one moment, Ian, just saying, what does the status of probability theory at the moment? Is it moving to explain more, or is it every time it takes a step forward from what you're saying,

it seems to take a step back, say yes but no, but, isn't it really? Well, I think the probability theory comes into everything we do in our lives. I mean, every time we go out on the street, we're assessing risk. And I think that's somehow where the interest, I think, is now, in sort of how much probability applies in medical experiments. if you get a test back saying you're positive for a particular, I think we understand probably it's very counterintuitive and the move now is to explain to the general population,

hopefully through a program like this, that the way probability works, because it is very counterintuitive. It's been used in legal cases to put people behind bars when actually the probability should have been used the other way to free the person. So there's a famous case of Sally Clark, who was she had two children that died of cop death. and a witness came up

and a specialist witness came on and said well there's a one in 73 million chance that that could happen because they thought they were independent variables multiply the probability together but in fact you know if one child dies in the family maybe there's a greater chance that the next child would also die of cot death

so it's not an independent variable also what's the chance that a woman kills both her children there's also probably about one in 73 million so why have you gone with actually saying that she it was killing the child rather than two cot deaths So this is a case where probability was used to put somebody behind bars and actually it should have been used to free them and eventually a witness came forward and said, you know,

a statistician came forward and explained the probability. So it's incredibly relevant to our everyday life in everything that we do. And still I don't think people, it's so counterintuitive quite often that we trick ourselves into things that shouldn't actually be interpreted like that. Well, I have to apologise for the Traits of Descriptions Act because I'm violating it, because I thought we'd get onto chaos theory, but we didn't. I'm not regarding that as a comment on the work for run the programme. You could say that. Anyway, the books of these three persons are available.

I'm sure they'll tell you all about chaos theory in their books. Thank you very much, Marcus de Sotoi, Kloveroni-Dougal, and Ian Stewart. And next week we'll be discussing the Russian geneticist, Lysenko, an intellectual life under Joseph Stalin. Thank you very much for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com.com.uk forward slash radio four.