The game
A fair coin is tossed repeatedly until it first lands tails. If the first tails appears on toss number k, the player receives 2^k units of money. So:
- Tails on the first toss pays 2 units.
- Tails on the second toss (heads, then tails) pays 4 units.
- Tails on the third toss pays 8 units.
- And so on, doubling each time.
The question is how much someone should be willing to pay to play. The problem was posed in the early 1700s by Nicolaus Bernoulli, and it takes its name from a paper by Daniel Bernoulli published in 1738 in the journal of the St Petersburg academy. It is a classic example of a gap between a mathematical average and what people sensibly do.
The expected value
Expected value is the sum of each possible payout multiplied by its probability. Tails first appears on toss k with probability (1/2)^k, because it takes k − 1 heads followed by a tail. The contribution of each case is:
- k = 1: probability 1/2, payout 2, contribution 1.
- k = 2: probability 1/4, payout 4, contribution 1.
- k = 3: probability 1/8, payout 8, contribution 1.
- And in general: (1/2)^k × 2^k = 1.
Every possible outcome contributes exactly 1 unit to the sum. There are infinitely many outcomes, so the sum is 1 + 1 + 1 + … without end. The expected payout is infinite. By the plain rule that a fair price equals expected value, a player should pay any finite amount to play.
Why nobody would
Almost nobody would pay a large amount. Look at what usually happens. Half the time the game ends on the first toss and pays 2. Three quarters of the time it pays 4 or less. Only one game in 16 pays 32 or more, since that needs the first four tosses all to be heads, which has probability (1/2)^4 = 1/16. The enormous payouts that make the average infinite are extremely rare.
The infinite average comes from a tiny chance of an enormous prize, and a single player will almost never see it. Over a few plays, the typical result is a small amount. The average only describes what happens over a huge number of plays, and even then the sample averages grow very slowly and erratically, because rare giant payouts dominate.
Resolution 1: the bank is not infinite
The game assumes the bank can pay any amount. No real bank, person or economy can. Suppose the most the bank can pay is 2^20, which is 1,048,576 units, so any game that lasts 20 or more tosses pays that capped amount.
Now compute the expected value again. Games ending on tosses 1 to 20 each contribute 1, for a total of 20. The remaining games, those where tails has not appeared in the first 20 tosses, have probability (1/2)^20 and pay the capped 2^20, contributing 1 more. The total is 21 units. With a bank limited to roughly a million units, the fair price is about 21, which feels much more sensible. The expected value grows only slowly as the cap rises: each doubling of the cap adds about 1 unit.
Resolution 2: money has diminishing value
Daniel Bernoulli’s own proposal was that the usefulness of money, its utility, grows more slowly than the amount. An extra unit means more to someone with little than to someone with a lot. If utility grows like the logarithm of wealth, the series of utility values adds up to a finite number, and a finite price results. This is the origin of the idea of risk aversion in economics.
This resolution has limits. If a game’s payouts were defined in utility rather than money, the paradox could be rebuilt with faster-growing payouts, so utility alone does not remove the issue in general. Different fixes address different parts of the puzzle.
Resolution 3: time and repeated play
A single play is also not the same as the long run. The law of large numbers says averages converge, but for this game the convergence is extremely slow, because the variance is infinite. A practical player cannot play the number of games needed to see the average. This is a reminder that expected value is a long-run idea, and that a single decision under risk involves more than the average.
What it teaches
The paradox shows that expected value is a useful tool but not a complete guide to decisions. A bet with a high or infinite average can still be a poor choice if the typical outcome is small, and the idea that a fair price equals the average depends on being able to play many times and survive the swings. That links to the bankroll maths of finite resources, and to variance, since the spread of outcomes matters as well as the mean.
It is also a caution about real gambling. Casino games have a house edge, which means a negative expected value for the player on each bet, and no clever framing of payouts changes this. Gambling can be harmful, outcomes are uncertain, and nobody should stake money they cannot afford to lose. Anyone who finds it hard to stop should contact a local support service.