Slopping thinking by the great mathematician has been enshrined for decades in the undergraduate economics and statistics curriculum. You know the story: the problem of whether to believe in the Christian god and follow church teachings can be formulated as one of expected value, the cost in pleasure of being religious times the likelihood that there is no such god compared with the cost in eternal damnation times the probability that the Bearded One actually exists. The small likelihood of divine existence is outweighed by the even greater imbalance in the value cofactor. Except that it’s all wrong.
There are different ways to approach the issue, but the simplest error is that of assuming only two possibilities, no god or the one particular god promulgated by Pascal’s church. Logically, the case for a Christian deity is no stronger than for many other pretenders. Pascal could follow all the church precepts, die and come face to face with a Hopi god, for example, or maybe a god no human group had latched onto. Perhaps the “godly” practices Pascal had adhered to would just annoy the true, ex post god. Or maybe, if there is a god, this omnipotent character has no self esteem problem to speak of and doesn’t care whether lowly mortals love, hate or ignore him/her/it. And I’m leaving out a wide swath of other possibilities, from multiple godheads to extra-dimensional beings we can’t begin to conceive of.
The point is that the expected value formula works only if you calculate over the full set of possible states or outcomes; the p’s have to add up to 1.
I’m reminded of this because of the column by Peter Bernstein in the Sunday NY Times, which repeats the Pascal chestnut even as it invokes the spirit (but not the name) of Nassim Nicholas Taleb to argue that we should anticipate the unexpected.