Petersburg paradox
/ˈpiːtəzbəːɡ ˌparədɒks/Mathematics
noun
Also St Petersburg paradox, Petersburg problem, St Petersburg problem. A paradox associated with certain betting games, for which calculation shows the expected winnings to be unlimited, but which are nevertheless unattractive to players because of the high probability of only a small payout.
- In a typical game, a coin is tossed until a head appears, the player winning one pound if a head appears on the first toss, two pounds if two tosses are required, four pounds if three tosses are required, and so on. The expectation of the player's winnings is calculated by multiplying the amount won if n tosses are required (2n−1) by the probability of winning that amount (1/2n) and taking the sum over all possible values of n. Since the multiplication gives a value of ½, and there are an infinite number of possible values of n, the expected winnings are infinite. Nevertheless most players would be deterred from paying to enter by the fact that the chance of the game ending on any throw is ½, and high payouts (more than the cost of entering the game) are very unlikely..
Origin
Mid 19th century; earliest use found in Isaac Todhunter (1820–1884), mathematician and historian of mathematics. From the name of St Petersburg (Russian Sankt-Peterburg), Russian city, home to a scientific academy in whose transactions the problem was first discussed by Daniel Bernoulli + paradox.