What are the chances of that?

Some months ago I was asked by the journal The Mathematical Intelligencer to review a recent popular science introduction to probability: Andrew Elliot's What are the Chances of that? (Oxford University Press, 2021). My review has now been published, and here is how it begins:

A stranger approaches you in a bar and offers a game. This situation recurs again and again in Andrew Elliot’s book What are the Chances of that? How to Think About Uncertainty, which attempts to explain probability and uncertainty to a broad audience. In one of the instances of the bar scene, the stranger asks how many pennies you have in your wallet. You have five, and the stranger goes on to explain the rules of the game. On each round, three fair dice are rolled, and if a total of 10 comes up you win a penny from the stranger, whereas if the total is 9 he wins a penny from you, while all other sums lead to no transaction. You then move on to the next round, and so on until one of you is out of pennies. Should you accept to play? To analyze the game, we first need to understand what happens in a single round. Of the 6³=216 equiprobable outcomes of the three dice, 25 of them result in a total of 9 while 27 of them result in a total of 10, so your expected gain from each round is (27-25)/216 = 0.009 pennies. But what does this mean for the game as a whole? More on this later.

The point of discussing games based on dice, coin tosses, roulette wheels and cards when introducing elementary probability is not that hazard games are a particularly important application of probability, but rather that they form an especially clean laboratory in which to perform calculations: we can quickly agree on the model assumptions on which to build the calculations. In coin tossing, for instance, the obvious approach is to work with a model where each coin toss, independently of all previous ones, comes up heads with probability 1/2 and tails with probability 1/2. This is not to say that the assumptions are literally true in real life (no coin is perfectly symmetric, and no croupier knows how to pick up the coin in a way that entirely erases the memory of earlier tosses), but they are sufficiently close to being true that it makes sense to use them as starting points for probability calculations.

The downside of such focus on hazard games is that it can give the misleading impression that mathematical modelling is easy and straightforward – misleading because in the messy real world such modelling is not so easy. This is why most professors, including myself, who teach a first (or even a second or a third) course on probability like to alternate between simple examples from the realm of games and more complicated real-world examples involving plane crashes, life expectancy tables, insurance policies, stock markets, clinical trials, traffic jams and the coincidence of encountering an old high school friend during your holiday in Greece. The modelling of these kinds of real-world phenomena is nearly always a more delicate matter than the subsequent step of doing the actual calculations.

Elliot, in his book, alternates similarly between games and the real world. I think this is a good choice, and the right way to teach probability and stochastic modelling regardless of...

Click here to read the full review!