Consider a hypothetical game in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a fixed amount initially, and then receives 2^n dollars if the coin comes up heads on the nth toss. The expectation value of the gain is then 1/2 × 2 + 1/4 × 4 + 1/8 × 8 + ... = 1 + 1 + 1 + ... = ∞ dollars. The Saint Petersburg paradox is that in this game, any finite amount of money can be wagered and the player will still come out ahead on average.

formulation date | 1713 (313 years ago) formulator | Nicolaus I Bernoulli proof date | 1738 (25 years later) (288 years ago) prover | Daniel Bernoulli

The paradox can be resolved by refining the decision model using the concept of marginal utility. The paradox can be resolved by taking into account the finite resources of the participants. The paradox can be resolved by noting that one cannot buy that which is not sold.

mathematical paradoxes