Alan Saunders: There's no getting away from probability. If you're worried about global warming, there'll be somebody out there to tell you the chances of a significant rise in temperature over the next 20 years.

Then there are the crime rates, the probability of being knocked down by a drunk driver. There's no end of it. Probability enters into statistics, physics, biology, chemistry, computer science and meteorology, and that's what we're looking at this week.

Hi, welcome to The Philosopher's Zone, I'm Alan Saunders.

In the 18th century, the English philosopher, Bishop Butler, said that probability is the very guide of life, and if that was true then, it's even more true now. Our lives are fraught with uncertainty, and probability is the study of uncertainty.

But what exactly are we doing when we're attributing a probability, low or high, to an event? To find out what probability is, I spoke to Alan Hájek, Professor of Philosophy at the Research School of Social Sciences at the Australian National University. And we began in the world of gambling; a good place to begin, given the national obsession and given that the Productivity Commission has just completed its report to the Federal government on gambling. And as I said to Alan Hájek, if we want to look at the development of ideas about probability, we have to go to the gaming house, don't we?

Alan Hájek: Absolutely. So we begin in the world of 17th century France around the 1650s, and gambling I guess was all the rage then, and there was a particular gambler, he must have been particularly good at it, the Chevalier de Mere, who was interested in a couple of games that he used to play. One of them you toss a fair die four times, you could either bet for or against a six coming up. Another game, you toss a pair of dice 24 times, and you wonder whether a double six will show up. And he noticed that the probability of winning in the first case seemed to be different from in the second case, but he wasn't quite sure why. So this was a gambling problem, and he turned to Pascal and Fermat, in their famous correspondence, they initiated probability theory.

Alan Saunders: So this is Fermat, the great mathematician noted for what's always referred to, I think probably inaccurately, as his 'Last Theorem,' and Blaise Pascal the great mathematician, philosopher, Christian thinker?

Alan Hájek: Absolutely, that's right. So they initiated probability theory with their correspondence, exactly as you say, by first analysing these gambling games, and then generalising to life more generally.

Alan Saunders: Well if we turn to gambling, and let's suppose I'm at a casino. There are 36 pockets in a roulette wheel, and the ball can go into only one of them at a time, so if I want to express the probability of getting any particular outcome, don't I just draw a line, write under it the number 36, which is the number of equally possible outcomes, and above it I write the number 1, because there's only one outcome that I'm interested in. This gives me a probability of 1 in 36. Now that seems pretty simple, doesn't it?

Alan Hájek: That's right. So this is the so-called classical interpretation of probability, so called because of its early and august pedigree with the likes of Pascal and Fermat. So that's right, you divide a problem into the number of possibilities and then probability just becomes a matter of counting the possibilities. You look at the number of favourable cases, you divide by the total number of cases, and that ratio is the probability.

Alan Saunders: There is of course a problem here, that I can do it that way, or I can say I want a favourable probability, say I want it to come up No.5, and either it comes up 5 or it doesn't, so it comes up 5 or it comes up not 5; that gives me two possibilities, and therefore the odds are 1 in 2.

Alan Hájek: Absolutely. So this is one of the famous problems with the classical interpretation, so what are these possibilities that we're just going to count over? So Laplace put it in the slightly weasely way. He said -

Alan Saunders: This is the 18th century French thinker and mathematician.

Alan Hájek: Yes, that's right. And he said the cases have to be equally possible, which seems like a sneaky way of saying equally probable, but yes, we have to be careful how we divide the space of possibilities in the first place.

Alan Saunders: While we're talking of gambling, there's something known in the business as the Monte Carlo fallacy. Tell us about that.

Alan Hájek: Or otherwise known as the gamblers' fallacy. OK, so in fact I've witnessed these very fallacy committed in Monte Carlo in a casino. So here was the case: a guy was betting on roulette and he kept putting money on No. 36, and then the wheel would spin, something other than 36 came up. He put more money on 36, again 36 did not come up. Still more money on 36. And I asked him in the end, 'Why do you keep putting money on 36?' and he looked at me like the answer should be obvious. He said, '36 has not come up all evening. It's due.' So that's an instance of the gamblers' fallacy, and more generally the fallacy is the thought that when you have some gambling device, some chance device that's produced a run of one outcome, then somehow chance is a self-correcting process and it's more likely that some other outcome will come up.

So it's a curious sort of bit of reasoning when you think about it. For a start, it's as if you're attributing a memory to the roulette wheel. It's as if the roulette wheel remembers 'Gee, I haven't come up 36 lately'. And secondly, it's as if you're attributing a conscience to the roulette wheel. 'Oh, I'd better do something about that, I'd better start tilting towards 36, and help it out a bit', as if 36 would suddenly become more likely. So that reasoning is what we call the gamblers' fallacy.

Alan Saunders: OK, well let me put in a word for the gambler here. The likelihood that 36 will not show up in say 36 spins of the wheel, is one assumes, relatively low. But the likelihood that it will not show up in 36 plus 1 is even smaller. Therefore, given that there's a long run of spins of the wheel in which 36 doesn't turn up, isn't it reasonable of me to assume that it's likely that it will show up next spin.

Alan Hájek: Well it's not reasonable to think that it's any more likely because of what it's done in the past. So in the long run, the outcome should appear roughly as one would expect them to, but not because chance is a self-correcting process, rather it's because the effects of particular runs gets diluted in the long run.

Alan Saunders: Now we've talked about some of the shortcomings of classical probability; an alternative to classical probability is the logical interpretation of the probability calculus; what is that?

Alan Hájek: Classical probability allegedly told us we could assign probabilities when we completely lacked evidence, or when, if we had evidence, it bore symmetrically on the possibilities, but now we generalise that. Logical probability codifies just relations of evidential support quite generally, so even if we have evidence and it bears asymmetrically on the possibilities, still we can put a number to the degree of support that the evidence bestows on a hypothesis.

Alan Saunders: Where do we get the number?

Alan Hájek: Right, well this was exactly the problem that Ramsey raised to Keynes.

Alan Saunders: This is Frank Plumpton Ramsey, the early 20th century, tragically short-lived Cambridge philosopher.

Alan Hájek: Exactly. Died at 26.

Alan Saunders: And he's talking to John Maynard Keynes, the great economist.

Alan Hájek: Exactly right. And they're talking in Cambridge, they were contemporaries there. So Keynes was a proponent of logical probability. He thought there really were these numbers that codified evidential support, you know, E bestows probability, P on proposition H, the hypothesis. Now Ramsey was sceptical of that, he could not discern these numbers, and this led him to be a pioneer of what we now call the subjective interpretation of probability; that we have rational degrees of belief, OK, we believe things to different degrees of confidence. If we're rational these are well-behaved, in fact they obey the probability calculus, but that's all that is to be said. We have rational agents assigning probabilities to things, they learn, they update their probabilities, but don't think that there's some objective, logical probability out there.

Alan Saunders: But we can't use rationality as part of our definition of the probability calculus can we, because you're saying that a rational person is somebody who acts in accordance with the probability calculus. So that's circular.

Alan Hájek: So the thought is a rational person is someone who has beliefs and desires, and they act in such a way as to further their desires in accordance with their beliefs. And now comes the substantive thesis: the way you do that is among other things, to obey probability theory. In fact Ramsay again in this classic piece of work, he had one of the most important arguments for obeying probability theory, the so-called Dutch Book argument. So the idea is we identify your subjective probabilities with your betting prices, so what's my confidence that it will rain in Sydney tomorrow? Let's say 30%. So that corresponds to my paying 30 cents for a bet, which pays a dollar if it rains, and nothing otherwise, and I'm indifferent between buying and selling the bet at 30 cents. OK? So we identify probabilities with betting prices, and then Ramsey argued if you don't obey probability theory with these degrees of belief, you would accept a set of bets each of which would look acceptable to you, would look reasonable, but put them together, and you'd be guaranteed to lose money, come what may. So however things turn out with the weather, I would lose money on these bets if I didn't obey probability theory.

So this was an argument for why you should obey probability theory, but this was not smuggling in some notion of rationality that was already probabilistic, it was just the idea that you want to serve your desires in accordance with your beliefs; sure losses of money would not serve your desires.

Alan Saunders: But we're also back in the world of Pascal and decision theory, aren't we, inasmuch as if I get a huge outcome for what seems like a relatively small outlay, or alternatively, if I'm just very rich and I can afford to be careless or reckless, I'm going to behave differently.

Alan Hájek: Yes, absolutely. So this is one of the problems with this whole interpretation of degrees of belief. So we're identifying them with betting prices, but there are all sorts of ways that betting prices could come apart from your true credences. So one way might be if you're very rich and you just don't care much about losing money; maybe you're the owner of a football team and you ostentatiously, publicly bet, much too favourable odds on your team's winning, not because that's what you really think but this is just your show of bravado and support of your team.

So we have various ways and reasons to misrepresent our true opinions in our betting prices. We can imagine a Zen Buddhist monk who's contemplating the Himalayas before him. You ask him, 'How much would you bet that that's Mount Everest before you?' and he'd say 'Go away, I'm meditating', he's just not even interested in participating in this gambling enterprise. Or you could imagine someone who likes gambling a little bit too much and is prepared to pay too much for a gamble. We often do that with lottery tickets for example, we get a kick out of the gambling itself, so again, we're not really measuring our true degree of belief.

Alan Saunders: Well I'd love to stay on gambling, because it's always fun, and there's a big Productivity Commission report on gambling, long awaited, coming out soon. I don't know whether that's going to avail itself of findings in probability theory, but let's go on instead to another alternative interpretation of probability, which is called the frequency interpretation. Now you I know have your doubts about this, but you also recognise that it's a bit like motherhood and apple pie. I mean a lot of people really do like it, don't they? So tell us what it is.

Alan Hájek: Yes, so this is maybe still the dominant interpretation of probability, especially among scientists. So it starts with a very natural thought. Look, probabilities are out there in the world, they're not just measures of our uncertainty. Quantum mechanics doesn't care about what we think, radium atoms decay with the half-lives that they do, independent of our thought. So we try to capture this notion, let's call it objective chance, something that's out there in the world, and the frequentist says, Well that's easy, you just conduct a number of trials of the relevant experiment, and just count how many times the event in question happens. If I want to know the probability that this coin has of landing heads, I toss the coin. I count the number of heads, divide it by the total number of tosses, bingo, that's the probability, according to the frequentist.

Alan Saunders: The problem there is it has difficulties when it comes to assigning probabilities to individual events. Now you can assign a probability to the toss of a coin, but only as one toss in a class of tosses of the coin, and there are circumstances where we want to ascribe a probability to an individual event. Now this was a problem that exercised the great 20th century philosopher, Karl Popper. So tell us what his solution was?

Alan Hájek: Absolutely. So this is the so-called problem of the single case. Again, in the case of a coin, if we toss the coin just once and then destroy it, count the number of heads, well it was either zero or one, and so the relative frequency is either zero or one, and we don't want to be committed to saying the probability is zero or one, in fact let's assume it was a fair coin, 50-50. So Popper developed what was called the propensity interpretation of probability, specifically designed to handle such single cases. In fact he was inspired by quantum mechanics.

And so the idea is that probability (well now we use again sort of weasel words like 'tendency' or 'disposition') but he's trying to convey the idea that these physical systems have certain tendencies that come in degrees to turn out certain ways. So in the context of a random experiment, propensity measures the tendency of one outcome or another. So we might say that the fair coin when tossed, in a nice fair way, has a tendency, he'd say propensity, of a half of landing heads.

Alan Saunders: And one of the questions he's addressing here is if you're tossing coins and you have a long row of tossings of the coin, and then into that row you inject a toss of a coin which is biased, which is always going to come down one side, the frequency theory doesn't give you a way of dealing with that, it just averages out over the total.

Alan Hájek: Yes. And the frequency theory always has to relativise its frequencies to some, as we say, reference class, so you know, it's this set of coin tosses. This gives rise by the way to another of the classic problems with frequentism, the so-called reference class problem. As an example, I care about my probability of living say to the age of 80, I care about that value. Well the frequentist says that we have to look at a reference class of people somehow like me, and we count how many of them live to 80, and we divide by the total number of such people. The trouble is you can put me in all sorts of reference classes, you know, I guess the reference class of middle-aged guys who exercise, not nearly enough, but also how about I'm a Woody Allen fan, you could put me in (at least the early movies) you could put me in -

Alan Saunders: The early funny ones.

Alan Hájek: The early funny ones - you could put me in the reference class of fans of the early Woody Allen movies, and so on. So it seems the only person who has all of my attributes is me. That's a pretty trivial reference class, and we're back to the problem of the single case.

Alan Saunders: OK. Well just finally. I got home this week and stuffed rather inexpertly into my mail box was something from the Federal government which was urging me to have a test, and I assume everybody's got this, it wasn't just a reference class of me - urging me to have a test for bowel cancer. And when anybody asks me to have a test for cancer, I immediately start asking myself probabilistic questions What are the likelihoods of false positives, what are the consequences of false positives, is it going to be an extremely unpleasant process to be treated for something I haven't got? So all these questions which are essentially probabilistic come to mind when I'm presented with something like that. So as I said at the beginning, probability is all around us. Now I'm not going to ask you whether you're being tested for bowel cancer, but ultimately where do you stand when it comes to interpreting probability?

Alan Hájek: Well let's take a test like that. So suppose that the test for bowel cancer is, let's say 99% effective. And what I mean by that is the probability given that you have the disease, that the test shows up positive for you is 99%, and also the probability given that you don't have the disease, of it showing up negative, that's 99%. Sounds like a pretty effective test. It seems like it's very reliable.

OK. Suppose you take the test, it comes out positive. How worried should you be? So offhand it seems the answer is 'very worried', it sounds like well 99% is the reliability of the test, that's the probability that you've got bowel cancer. Actually the proper answer is I just haven't given you enough information. I need to tell you the so-called base rate, I need to tell you what's the probability antecedently that you would have this disease? So we imagine looking at the whole population and we ask what's the prior probability that you would have bowel cancer? So so-called Bayes Theorem tells us that you need to pay attention to this prior probability as well.

Alan Saunders: We should just say Bayes, this is an 18th century English guy who came up with a theorem in probability theory which is actually quite fashionable amongst philosophers today.

Alan Hájek: It's all the rage. So much so that a whole movement called Bayesianism is named after him. I actually find that slightly surprising though - this might annoy some people but Bayes' Theorem is just one of many theorems in probability theory. It's usually associated with subjective probabilities, you know, this interpretation of probabilities called Bayesianism, which is funny, because it's just a theorem of probability that's neutral as to how you interpret it. Anyway it's just one of many theorems, it's a nice theorem, it's one way to calculate conditional probabilities. It's not the only way.

Alan Saunders: But when you talk about the bowel cancer issue, you talk about base rates which I think implies a pull on your part towards the frequency interpretation, would that be right?

Alan Hájek: Well frequencies are often good information and so I would certainly pay attention to the frequency information. They would inform my own subjective probabilities. Other things might, too. So certainly in the case we're talking about, it does seem pretty relevant that there's a certain prior probability in the population for having the disease. So just to consider some cases: suppose that say 1% of the population has the disease. So how worried should you be when you turn out to have a positive test, and remember that the test was 99% reliable in the sense that I gave. It turns out the answer is a half, so it's not 99%, it's a half is the measure of how worried you should be. Suppose let's say 1 in 1,000 people have the disease, then it turns out the correct answer is .09, that's the probability that you have the disease, given your positive test.

So in that sense, it's a sort of comforting thought that it's antecedently improbable that you would have this disease, which is not to say you should suddenly stop taking the bowel cancer tests, this would be bad advice, partly because of the stakes involved. But Bayes' Theorem does tell you how you have to temper this stuff about the reliability of the test with the antecedent probability of having the disease.

Alan Saunders: Well probability, as we said at the beginning, is the guide of life. I'm not sure that I feel guided towards having a test or not, but it was a fascinating discussion. Alan Hájek, thank you very much indeed for joining us.

Alan Hájek: I thought it would be a good bet to come on this show, and I was right. Thanks very much.

Alan Saunders: And if you want to engage in conversation about probability, just go to our website.

The show is produced by Kyla Slaven, with technical production by Charlie McKune. I'm Alan Saunders, I'll probably be back next week.