Drunkard's Walk - Leonard Mlodinow [32]
Galileo employed his scientific skills to write a short piece on gambling, “Thoughts about Dice Games.” The work was produced at the behest of his patron, the grand duke of Tuscany. The problem that bothered the grand duke was this: when you throw three dice, why does the number 10 appear more frequently than the number 9? The excess of 10s is only about 8 percent, and neither 10 nor 9 comes up very often, so the fact that the grand duke played enough to notice the small difference means he probably needed a good twelve-step program more than he needed Galileo. For whatever reason, Galileo was not keen to work on the problem and grumbled about it. But like any consultant who wants to stay employed, he kept his grumbling low-key and did his job.
If you throw a single die, the chances of any number in particular coming up are 1 in 6. But if you throw two dice, the chances of different totals are no longer equal. For example, there is a 1 in 36 chance of the dice totaling 2 but twice that chance of their totaling 3. The reason is that a total of 2 can be obtained in only 1 way, by tossing two 1s, but a total of 3 can be obtained in 2 ways, by tossing a 1 and then a 2 or a 2 and then a 1. That brings us to the next big step in understanding random processes, which is the subject of this chapter: the development of systematic methods for analyzing the number of ways in which events can happen.
THE KEY TO UNDERSTANDING the grand duke’s confusion is to approach the problem as if you were a Talmudic scholar: rather than attempting to explain why 10 comes up more frequently than 9, we ask, why shouldn’t 10 come up more frequently than 9? It turns out there is a tempting reason to believe that the dice should sum to 10 and 9 with equal frequency: both 10 and 9 can be constructed in 6 ways from the throw of three dice. For 9 we can write those ways as (621), (531), (522), (441), (432), and (333). For 10 they are (631), (622), (541), (532), (442), and (433). According to Cardano’s law of the sample space, the probability of obtaining a favorable outcome is equal to the proportion of outcomes that are favorable. A sum of 9 and 10 can be constructed in the same number of ways. So why is one more probable than the other?
The reason is that, as I’ve said, the law of the sample space in its original form applies only to outcomes that are equally probable, and the combinations listed above are not. For instance, the outcome (631)—that is, throwing a 6, a 3, and a 1—is 6 times more likely than the outcome (333) because although there is only 1 way you can throw three 3s, there are 6 ways you can throw a 6, a 3, and a 1: you can throw a 6 first, then a 3, and then a 1, or you can throw a 1 first, then a 3, then a 6, and so on. Let’s represent an outcome in which we are keeping track of the order of throws by a triplet of numbers separated by commas. Then the short way of saying what we just said is that the outcome (631) consists of the possibilities (1,3,6), (1,6,3), (3,1,6), (3,6,1), (6,1,3), and (6,3,1), whereas the outcome (333) consists only of (3,3,3). Once we’ve made this decomposition, we can see that the outcomes are equally probable and we can apply the law. Since there are 27 ways of rolling a 10 with three dice but only 25 ways to get a total of 9, Galileo concluded that with three dice, rolling a 10 was 27/25, or about 1.08, times more likely.
In solving the problem, Galileo implicitly employed our next important principle: The chances of an event depend on the number of ways in which it can occur.