The St. Petersburg Paradox

The problem
flip until tails · win 2ⁿ if tails on flip nHflip 1$2Hflip 2$4Hflip 3$8Hflip 4$16…how much would you pay to play?

A casino offers a game: a fair coin is flipped until it lands tails. The payout depends on which flip produced the first tails:

First tails on flip nnProbabilityPayout
112\tfrac{1}{2}\2$
214\tfrac{1}{4}\4$
318\tfrac{1}{8}\8$
nn12n\tfrac{1}{2^n}\2^n$

How much should you be willing to pay to play this game once?

Tempting (but wrong)
most people pay $5 – $20~ $10 ?feels like a 50/50 game, mostly small winsactual expected value: infinity

The instinctive answer is in the \5toto$20$ range. The reasoning: a fair coin lands tails fast, so most rounds end with a small payout. You'd "probably win 4 or 8 bucks," so pay something in that ballpark.

This treats the game like it has a reasonable typical outcome. The math disagrees in a spectacular way.