Whenever I hear about "Pascal's Wager", the phrase "worst philosophical argument ever" inevitably springs to mind. One of the main reasons for this close association is described here (in short, the argument idiotically assumes that Christianity and atheism together exhaust the theological possibilities). But there's also another problem with it, which I mentioned only briefly in that past post, and that is the following bit of reasoning:
Since Expected utility = Reward * Probability, and infinity multiplied by any nonzero probability is still infinity, we conclude that there's an infinite expected utility to believing in God [assuming that there is some non-zero probability (no matter how small!) of receiving an infinite reward for doing so].
Now, as Alan Hajek neatly pointed out, this argument "proves too much". For suppose I decide to flip a coin, and will believe in God only if it lands on heads. This process also has a non-zero chance of obtaining infinite utility (by the Pascalian assumption). So the expected utility of this course of action is, again, infinite. Or suppose I decide to believe in God only if I win the lottery next week. Again, non-zero probability => infinite expected value. Because infinity is involved, the lower probabilities make no difference at all. The maths implies that it would be just as rational to choose a one-in-a-million chance of infinite reward as to choose a guaranteed infinite reward. But that's clearly absurd!
Getting back to Pascal: whatever I do, there is presumably some (miniscule, but non-zero) chance that I'll end up believing in God because of it. So, if Pascal's wager is sound, we've just proved that going down to the pub for a beer has infinite expected utility. A drink sounds pretty good right about now. ;)