The St. Petersburg paradox

Question

A fair coin is tossed until the first head appears. If the first head comes on toss n, you are paid 2^{n} dollars. What is the expected payout of this game, and how much should you pay to play it?

Accepted Answer

The first head lands on toss n with probability \left(	frac12ight)^n, and the payout is 2^n. So each term of the expectation is P(n)\cdot 2^n = \frac{1}{2^n}\cdot 2^n = 1. Therefore E[	ext{payout}] = \sum{n=1}^{\infty} 1 = \infty. The expected value is infinite, yet almost no one would pay even 20 to play. That tension is the paradox: expected value alone gives a nonsensical price. The payout is \2 with probability 	frac12, \4 with probability 	frac14, and so on; you collect a modest amount with overwhelming probability, and the infinite mean is driven entirely by astronomically rare, a…

The St. Petersburg paradox

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