The game with infinite expected value

Question

A casino offers this game. A fair coin is flipped until it first lands heads. If that happens on flip number n, you are paid 2 to the power n dollars: 2 if heads comes first flip, 4 if on the second, 8 on the third, and so on. 1. What is the expected payout? 2. Almost nobody would pay more than about 20 dollars to play. Explain why the expected value is a poor guide here, and give at least two different resolutions.

Accepted Answer

The expected payout is infinite. Yet a sensible price is modest, and the gap is the whole point. The expectation. The first heads is on flip n with probability 1 / 2 to the n, and the payout is then 2 to the n. Each term of the expected value is (1 / 2 to the n) × 2 to the n = 1, and there are infinitely many terms: 1 + 1 + 1 + ... = infinity. Why nobody pays much. Look at the actual distribution: half the time you win 2 dollars, a quarter of the time 4, an eighth of the time 8. The chance of winning more than 1,000 dollars is under 1 in 1,000, and the chance of more than a million is under o…

The game with infinite expected value

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