Problem 11

Daniel Bernoulli and the St. Petersburg Problem (1738)

Problem. A player plays a coin-tossing game with a fair coin in a casino. The casino agrees to pay the player 1 dollar if heads appears on the initial throw, 2 dollars if head first appears on the second throw, and in general 2^{n−1} dollars if heads first appears on the nth throw. How much should the player theoretically give the casino as an initial down-payment if the game is to be fair (i.e., the expected profit of the casino or player is zero)?

Solution. The player wins on the nth throw if all previous (n − 1) throws are tails and the nth thrown is a head. This occurs with probability and the player is then paid 2^{n−1} dollars by the casino. The casino is therefore expected to pay the player the amount

Thus it seems that no matter how large an amount the player initially pays the casino, she will always emerge with a profit. Theoretically, this means that only if the player initially pays the casino an infinitely large sum will the game be fair.

# 11.1 Discussion

This problem has been undoubtedly one of the most discussed in the history of probability and statistics. The problem was first proposed by Nicholas Bernoulli (1687–1759) in a letter to Pierre Rémond de Montmort (1678–1719) in 1713.^{1} It was published in Montmort's ...