Infinite mean, but what's the typical win?

Question

In the St. Petersburg game a fair coin is tossed until the first head; if it appears on toss $n$ you are paid $\$ 2^n$. The expected payout is infinite.

What is the median payout, and what is the probability of winning more than $\$ 1000$? Contrast these with the infinite mean.

Accepted Answer

The payout distribution is: \2 with probability 	frac12, \4 with probability 	frac14, \8 with probability 	frac18, and so on (\2^n with probability 2^{-n}). Median. The median is the payout you get at least half the time. Since the very first outcome, a head on toss 1, pays \2 and occurs with probability 	frac12, at least half of all plays pay \2 or less. In fact exactly 	frac12 of plays pay exactly \2. So the 	ext{median payout} = \2. The single most likely outcome, and the typical one, is the smallest possible prize. Probability of winning more than \1000. You need 2^n 1000. Since 2^9…

Infinite mean, but what's the typical win?

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