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the unfinished game
19 april 2009
Let's say the Cubs are leading the Sox in the World Series, two games to one. Let's also say that any Cubs-Sox game is a toss-up; the teams are so evenly matched that you might as well be flipping a coin. How likely are the Cubs to win that World Series?
- God only knows.
- 1-1. The two teams are evenly matched, aren't they?
- 2-1, because the Cubs are up 2-1.
- 3-2. There are five possible ways for the Series to end: Cubs in 5, 6, or 7 games; Sox in 6 or 7. The Cubs win in three of those five possible outcomes.
- 5-3. There are ten possible paths to those five outcomes. The Cubs win 6 of those and the Sox win 4.
- 11-5. There are sixteen possible outcomes if they kept on playing after one team had won four games, until seven games had been completed in each case.
If you're confused, you're in good company. The scenario I've just presented is an "unfinished game," the topic of Keith Devlin's recent book. In 1654, the great thinkers Pascal and Fermat conducted a correspondence on unfinished games. They were confused too. Or at least Pascal was. By the end of their correspondence, Fermat had used Pascal's constructive confusion to establish a theory of the unfinished game that was to become one basis for the modern science of statistics. Or, as Devlin's subtitle would have it, Fermat "made the world modern."
But that's just subtitle talk. I swear, every popular history book you pick up nowadays is about some sine qua non that made the world modern, changed it indelibly, set it on its unalterable course towards us. Split, Toast, Spread!: How Thomas's Invented Nooks and Crannies and Changed Breakfast Forever. Velcro: How a Little-Heralded Sticker-Together Upset the Fastener World and Created the Globalized Economy. Splish Splash: How Intermittent Windshield Wipers Ushered in a Whole New Era of Transportation. Give me a break.
But I digress. The correct answer to the Cubs-Sox series problem, incidentally, is 11-5. If this situation occurs annually, over time the Cubs will win 11 of every 16 Series. The usefulness of the 11-5 solution is that it suggests how to figure betting odds. Once we know how to figure betting odds, we can calculate all kinds of risks: actuarial risks for insurers, financial risks for options markets, player-projection risks for fantasy baseball players, and other elements of modernity.
Pascal, incidentally, was not confused about the 11-5 solution of the two-player unfinished game. He got that rather quickly. Where he got out of his depth was in a much more complicated three-player problem that fills the middle of a famous letter to Fermat, and fills the middle of Devlin's book. That's where I got out of my depth too: "Le silence éternel de ces espaces infinis m'effraye," as Pascal might say. But Fermat, Devlin assures me, was way ahead of his pensée-producing pal.
The basic form of my problem with the three-player unfinished game is invincible denseness. But the form of Pascal's problem seems to have been uncertainty over how to consider those situations where the series, IRL, would not continue. Those who pick 3-2, or 5-3 as the correct odds in the Cubs-Sox series generally make the mistake of thinking that a series where the Cubs win 4 games to 1 represents just one possibility. In fact it represents one quarter – 4/16 – of all the possibilities, not one-fifth or one-tenth.
Devlin presents another puzzler as an analogue, the famous kids'-gender enigma that occasionally shows up in airline magazines to infuriate passengers. If I have two kids, the chance that one's a boy and one's a girl is 1 in 2. The chance of two boys is 1 in 4, as is the chance of two girls. (That's because there are four ways two children can get born: boy-boy, boy-girl, girl-boy, girl-girl.)
So let's say I have two kids and I tell you that one is a girl. What are the odds that the other kid will be a boy? They are in fact 2 in 3, though the solution seems highly counterintuitive. But by telling you I have a girl, I remove the "two boys" outcome from play. Four outcomes (boy-boy, boy-girl, girl-boy, girl-girl) have been reduced to three; two of those outcomes include a brother for my daughter.
Dang. What continues to perplex many airline passengers, and not a few mathematicians, is that if I'd told you that the older of my two kids was a girl, the odds of the younger being a boy would drop back down to 1 in 2. (I've removed both "boy-boy" and "boy-girl" from the matrix of outcomes.) Much of Devlin's interesting discussion in The Unfinished Game centers on problems where initial odds are complicated by subsequent knowledge. Such problems have applications everywhere from poker to stimulus bills.
As someone who has talked many a freshman critical-thinking class through the Monty Hall Problem, I dig these puzzlers when they are at or below the Cubs-Sox level of complexity. Anything beyond that and I want to curl up and emit small bleating noises. Devlin's Unfinished Game shows vividly that even the simpler realms of probabilty can tax the average human brain. The world has not become modern because a lot of us have gotten better at these calculations. It's gotten modern because the occasional Fermat has parlayed advances in logic into probabilistic industries. And as the behavior of the financial markets in the past year has taught us, modern man with modern computers is sometimes not much better than Blaise Pascal and his desktop calculator at assessing risk.
Devlin, Keith. The Unfinished Game: Pascal, Fermat, and the seventeenth-century letter that made the world modern. New York: Basic Books, 2008.