# THE DEVIL’S OFFER3

Suppose Ms C dies and goes to hell, or to a place that seems like hell. The devil approaches and offers to play a game of chance. If she wins, she can go to heaven. If she loses, she will stay in hell forever; there is no second chance to play the game. If Ms C plays today, she has a 1/2 chance of winning. Tomorrow the probability will be 2/3. Then 3/4, 4/5, 5/6, etc., with no end to the series. Thus, every passing day increases her chances of winning. At what point should she play the game?

The answer is not obvious: after any given number of days spent waiting, it will still be possible to improve her chances by waiting yet another day. And any increase in the probability of winning a game with infinite stakes has an infinite utility. For example, if she waits a year, her probability of winning the game would be approximately 0.997268; if she waits one more day, the proba- bility would increase to 0.997275, a difference of only 0.000007. Yet even 0.000007 multiplied by infinity is infinite.

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On the other hand, it seems reasonable to suppose the cost of delaying for a day to be finite—a day’s more suffering in hell. So the infinite expected ben- efit from a delay will always exceed the cost.

This logic might suggest that Ms C should wait forever, but clearly such a strategy would be self-defeating: why should she stay forever in a place in or- der to increase her chances of leaving it? So the question remains: what should Ms C do?

5. Pascal is famous for invoking infinite utilities to argue for belief in God. Explain and evaluate his argument in the following passage.

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