The French mathematician Blaise Pascal (also a physicist, inventor of one of the first calculators, with his name given both to a unit of pressure and a computer programming language) put forward a proposition — known as Pascal’s Wager — that from a logical standpoint, it is safer to accept the existence of God. Of course, faith is another matter entirely, and Pascal himself was not so shallow as to embrace belief solely on the basis of this theory. On the contrary, alongside his mathematical and physical genius, he was also an important philosopher. While reading about this in The Skeptical Angel, I came across this link. There, the moral contradictions of this thesis are mentioned, as well as arguments regarding the possibility of multiple gods, and that a life with no beginning or end for humankind might be equivalent to eternity. Among these, one statistical objection concerning belief in God and eternal salvation caught my attention:
However, Duff (1986) and Hájek (2003) argue that the argument is in fact invalid. Their point is that there are strategies besides wagering for God that also have infinite expectation — namely, mixed strategies, whereby you do not wager for or against God outright, but rather choose which of these actions to perform on the basis of the outcome of some chance device. Consider the mixed strategy: “Toss a fair coin: heads, you wager for God; tails, you wager against God”. By Pascal's lights, with probability 1/2 your expectation will be infinite, and with probability 1/2 it will be finite. The expectation of the entire strategy is:
1/2*∞ + 1/2[f2p + f3(1 − p)] = ∞
That is, the ‘coin toss’ strategy has the same expectation as outright wagering for God. But the probability 1/2 was incidental to the result. Any mixed strategy that gives positive and finite probability to wagering for God will likewise have infinite expectation: “wager for God iff a fair die lands 6”, “wager for God iff your lottery ticket wins”, “wager for God iff a meteor quantum tunnels its way through the side of your house”, and so on.
In short, if we treat salvation as the outcome of a function, E[x], then even if infinity appears in the equation with the smallest fraction, the expected salvation still comes out as infinite.
I’ll close the topic by saying that the essence of religion is faith. But I made this long preamble because of the association it triggered in me, which led me to the following conclusion:
Let’s suppose that faith — belief in God — brings eternal salvation, while sins, good deeds, and acts of worship provide people with additional losses or gains (since a ledger will be handed over, after all). If person A’s faith is 0, then the expected salvation will be:
E[A] = 0 + x
And even if x is positive (if good deeds and worship outweigh sins), once x is exhausted, in the end he will still be condemned to hell.
On the other hand, even if there is the tiniest fragment of belief within him,
E[A] = 0.00001 * ∞ − y = ∞
his salvation will be eternal. No matter how negative y may be, salvation remains infinite.
Note: I use the words “faith” and “believer” because they correspond to the meaning here; they have nothing to do with robe-wearing clerics. A faithful person is a good person.