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this case, that your faith will save your soul—and A is the amount promised to winners. 2 E is expectation, what you could hope to gain for your stakes; and in this case, you bet your life.
    Pascal says that, since we have no way of knowing God, we can assume equal probabilities of winning and losing, so p ( X ) = 1/2. At these odds, you need to be offered only two lives to make the game fair; if three were offered, you’d be a fool not to bet on God’s side.
    But if faith actually does save your soul, there is “an infinity of an infinitely happy life to gain.” Since the payoff stands in relation to the stake as the infinite to the finite, you should always bet on God, whatever the odds against you: “There is nothing to decide—you must give everything.”
    The popular summary of Pascal’s Wager is “Bet on God—if He exists, you win; if He doesn’t, you don’t lose anything.” Pascal was not so cynical; for him, the calculation genuinely expressed a belief that probability can offer a handle on the unknown, even if that unknown were as great as the question of our salvation.
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    The Chevalier de Méré was not simply a high-living gamester but also a capable mathematician. The first problem he brought to Pascal was this: he knew that there is a slightly better than even chance of throwing at least one six in four throws of a die; adding a second die to the throw should simply multiply the number of possible outcomes by six—so, throwing two dice, shouldn’t there be a better-than-even chance of getting at least one double-six in 24 throws? But gamblers were noticing that double-six showed up slightly less than half the time in 24 throws. De Méré, Pascal wrote, “was so scandalized by this that he exclaimed that arithmetic contradicts itself.”
    Pascal, of course, could not accept such an insult to mathematics; his solution followed the path we have already explored with Cardano. The chance of throwing at least one double-six in 24 throws is the inverse of the chance of not throwing a double-six (whose probability is 35/36) in any of 24 throws; we can swiftly calculate it as:

    That is, a bit less than 1/2. Pascal, without mechanical help, had redeemed the accuracy of arithmetic: the shooter who bets on making a double-six will lose over time on 24 throws, and will win on 25—just as the gamblers had found.
    De Méré’s second problem, the “problem of points,” is deceptively simple. Let’s say you and a Venetian have put your stakes on the table; the first to win a certain number of games will pocket the lot. Unfortunately (and here it’s tempting to think of some Caravaggio painting of low-life suddenly interrupted by an angel), the game is stopped before either of you has reached the winning total. How should the pile of money on the table be divided?
    Fermat (of the Theorem), with whom Pascal discussed the problem, chose a method that adds the probabilities of mutually exclusive events. Let’s say the game is to throw a six in eight throws; you are about to roll the first time when—behold!—radiance fills the darkened tavern and we are called to higher things. But the money; we can’t just leave it there. Well, you could have made the point on your first roll; you had a 1/6 chance of that, so take 1/6 of the pot. But, alternatively, you could have failed on the first but made it on the second, so take 1/6 of the remainder, or 5/36 of the total. Then there’s the third throw—a further 25/216; and the fourth; add another 125/1296 . . . and so on eight times, adding probabilities and hoping at each stage that someone will have the correct change.
    Fermat was really interested in the problem only as a mathematical construct, but adding up cases of possible success, as Fermat does, can rapidly become a matter of argument: for instance, if you had made your point on the first throw, you