The St Petersburg Paradox

Consider a game: flip a fair coin repeatedly until it comes up tails. If the first tails happens on flip $n$, you win $2^n$ dollars. The expected payout is the sum, over every possible number of flips, of the probability of stopping there times the payout: each term contributes exactly $1 regardless of how long the game runs, and there are infinitely many possible outcomes, so the expected value of the game is infinite.

Yet almost no one would pay even $25 to play this game once, let alone an infinite amount. The paradox is that a game with literally infinite expected value has essentially no practical worth to a real player, because the huge payouts that drive the average up are also vanishingly unlikely — most games end in the first few flips for a tiny payout, and the enormous winnings that make the average infinite occur so rarely that virtually no one will ever see them in a lifetime of play.

Daniel Bernoulli's resolution was that people don't actually maximize expected monetary value — they maximize expected utility, and utility grows more slowly than money itself (diminishing marginal utility), so a diminishing-returns utility function like a logarithm turns the infinite expected payout into a small, finite, and sensible expected utility, matching what people are actually willing to pay.