1
Probability: A Two-Faced Guide to Life?

1    Why Care about Probabilities?

A book on how to understand probability may not sound interesting; in fact, it probably doesn't sound interesting if you're not interested in maths. But if you don't understand probability, then you'll probably find yourself making some bad decisions. (Maybe it would pique your interest if I told you that I made a lot of money, from people who didn't know as much about probability as they should have, during my student days? More on this in Chapter 3.) Sometimes you'll act when you shouldn't, and other times you'll fail to act when you should. Don't take my word for it. Let's think about scenarios in which claims involving probabilities are relevant in everyday life.

Imagine you're intent on climbing a mountain, and you consult the weather forecast for the day. On the report, you see that the probability of precipitation in the relevant mountain range - or what is sometimes called the chance of rain - is just one in twenty, or 5 per cent. Should you take waterproof gear with you?

Obviously this will depend a little bit on context, so let's fill some of that out. Imagine you don't have any waterproof gear, and that it will be quite a hassle to get some, but that you don't want to get wet. Overall, you think that getting wet would be more unpleasant than going through the trouble of getting the gear; in an ideal world, however, you'd neither get the gear nor get wet. It's possible to assign a number to each possible outcome, a utility, to make this kind of discussion more precise. Let's avoid complicating things unnecessarily, though. We can instead rank the four possible outcomes in order of your preference: no gear and no rain (best), gear and rain (2nd best), gear and no rain (3rd best), and no gear and rain (worst). (It's always helpful in such scenarios to think about whether anything has been assumed which hasn't been explicitly mentioned. I encourage you to do this throughout the book. In this case, for example, 'gear and rain' has been ranked more highly than 'gear and no rain'. I did this because I figured you'd be a bit irritated at having the gear if it didn't rain; you'd be thinking 'I shouldn't have bothered to get this gear!' But perhaps I should have added this as a stipulation in presenting the context.)

The order of preference makes it clearer what's at stake in this hypothetical scenario. If you take the gear, you miss out on the best possible outcome. But you also protect yourself from the worst possible outcome (while giving yourself a shot at the second and third best outcomes). Now if the order of preference were the only information you had, your choice might depend only on your attitude towards risk; some people are more risk-averse than others. But you also know that no rain is much more probable than rain, which may affect your decision. And indeed it should affect your decision, as we will soon see, when 'much more probable' is interpreted in some of the available ways. Very roughly, we may capture why by saying that probability is often used as a measure of the salience of various possibilities.

Maybe you are still unconvinced that probabilities, so construed, are important. So imagine that you failed to rank possibilities by salience. You would treat any possibility you identified the same as any other. You would treat the possibility of a meteorite landing on your head, or of being accosted by a knife-wielding psychopath, as seriously as you would the possibility of rain. You would be worried about whether to wear a hard hat, a stab-proof vest, and so forth. In fact, with a little imagination, you'd be worried about so many possible fates that you'd be overwhelmed and confused. (Of course, inaction can be bad too. If you stay in, you might die in an earthquake. And so on!) The only upside would be that you'd be able to consider lots of good possibilities as well as the bad ones; of stumbling on a hidden cache of diamonds, of meeting a future partner, and so forth. But really, you'd have no way of proceeding other than guessing about what was best to do. Life would be a series of such guesses. And most of us don't treat life that way, in so far as we think that anyone who wears a hard hat at all times is crazy.

However, this still leaves us with the question of what, exactly, we can and should understand probability talk to reflect. And this is the main question that this book focuses on. One way of thinking about the issue is as follows. How can we satisfactorily translate a claim like 'The probability of rain in Hong Kong today is 0.5' into a descriptive claim that does not involve a mention of probability? (Note the use of 'descriptive'. We don't want the claim just to be a statement about how one should act. We want it to provide a reason for acting in a particular way.) Whether such a statement should be used to guide one's actions - and which actions, if so - depends on how this is done.

We will come to this below. But beware that there are all kinds of little tricks and subtleties that are easy to miss if you don't think carefully about probability statements. Consider again, for instance, the scenario discussed above. You know that the probability of precipitation in the relevant mountain range is one in twenty. But you only expect to climb one mountain in that range. So might the probability of rain on the mountain you intend to climb (or better still, on the route you intend to take) be different from one in twenty? Might it be lower? Might it be higher? Have a think about this before you read on.

2    The Two Faces of Probability

Imagine that I take a normal coin out of my pocket. What do you think the probability is that I will get a result of 'heads' when I flip it? As it happens, I have asked this question to students on numerous occasions. The following dialogue illustrates how one of these discussions - the best one! - developed:

darrell: What is the probability that I'll get 'heads' when I flip this coin? student one: The probability is one half. darrell: Does anyone disagree? student two: I think we don't know what the probability is. darrell: Really? Why's that? student two: We would need to do an experiment to estimate the probability. darrell: So what would you say to Student One? student two: The coin - or your flipping of it - may be biased. We don't know if it is. So we have to do an experiment to find out if it is, and by how much if so. darrell: OK. So you think that if I flip the coin repeatedly, and we record the frequency of heads results, then we'll be in a better position to estimate the probability of a heads result on a flip? student two: Yes, that's right. darrell: So on your view, the real probability would definitely become apparent if, by some kind of magic, we carried on this process forever. That is, if I flipped the coin infinitely many times and we had all the results? student two: I suppose so, yes. darrell: OK. How would you respond to this, Student One? student one: Well it's true that the coin might be biased . . . darrell: . . . by which you mean that the process of me flipping the coin might turn out to give one result with a higher frequency than the other? student one: Sure, we can understand it that way. But even if we imagine that the coin is biased, we don't know which way it's biased. So it seems right to conclude that the probability of a heads result is the same as the probability of a tails result given what we know. darrell: OK. So this means that there's a real difference in your views. Correct me if I'm wrong, Student Two, but on your view the probability may be different if I swap this coin for another one? That is, because the experimental setup will change? student two: That's right. darrell: But the probability on your account will stay the same if I change the coin, Student One? student one: Yes, that's right. Well . . . it's right if you don't tell us anything more about - or I...