Flip a coin until the first tails. The payout is 2^(k-1) where k is the flip that lands tails. The expected value is infinite — yet nobody pays infinity. Why not?
Bernoulli's fix: a player with wealth w and logarithmic utility pays c so that E[ln(w − c + payout)] = ln(w). The answer is finite and grows only slowly with wealth — resolving the paradox.