When it comes to making bets, Pascal observed, it is not enough to know the odds of winning or losing. You need to know what’s at stake. You might want to take unfavorable odds if the payoff for winning would be really huge, for example. Or you might consider playing it safe by betting on a sure thing even if the payoff was small. But it wouldn’t seem wise to bet on a long shot if the payoff was going to be meager.
Pascal framed this issue in his religious writings, specifically in the context of making a wager about the existence of God. Choosing to believe in God was like making a bet, he said. If you believe in God, and that belief turns out to be wrong, you haven’t lost much. But if God does exist, believing wins you an eternity of heavenly happiness. Even if God is a low-probability deity, the payoff is so great (basically, infinite) that He’s a good bet anyway. “Let us weigh the gain and the loss in wagering that God is,” Pascal wrote. “Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.”2
Pascal’s reasoning may have been theologically simplistic, but it certainly was mathematically intriguing.3 It illustrated the kind of reasoning that goes into calculating the “mathematical expectation” of an economic decision—you multiply the probability of an outcome by the value of that outcome. The rational choice is the decision that computes to give the highest expected value. Pascal’s wager is often cited as the earliest example of a math-based approach to decision theory.
In real life, of course, people don’t always make their decisions simply by performing such calculations. And when your best decision depends on what other people are deciding, simple decision theory no longer applies—making the best bets becomes a problem in game theory. (Some experts would say decision theory is just a special case of game theory, in which one player plays the game against nature.) Still, probabilities and expected payoffs remain intertwined with game theory in a profound and complicated way.
For that matter, all of science is intertwined with probability theory in a profound way—it’s essential for the entire process of