Flip a coin until tails. Win $2^n where n = heads count. Expected value is infinite, but would you pay $100 to play?
Expected value = $1 + $1 + $1 + ... = infinity. Each term contributes $1: P(n heads) = 1/2^n, payout = $2^n, so each term = $1.
Yet the median payout is just $2! Half the time you flip tails immediately. The "infinite" expected value comes from astronomically rare but massive payouts.
Resolution: use log utility (Daniel Bernoulli, 1738). A rational agent values $1M much less than 1000x $1K.