Gambler's Ruin

Let two players each have a finite number of pennies (say, $n_1$ for player one and $n_2$ for player two). Now, flip one of the pennies (from either player), with each player having 50% probability of winning, and give the penny to the winner. If the process is repeated indefinitely, the probability that one or the other player will eventually lose all his pennies is unity. However, the chances that the individual players will be rendered penniless are

$$P_1 = {n_1\over n_1+n_2}$$

$$P_2 = {n_2\over n_1+n_2}.$$

See also Coin Tossing, Martingale, Saint Petersburg Paradox

References

Cover, T. M. ``Gambler's Ruin: A Random Walk on the Simplex.'' §5.4 in Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, p. 155, 1987.

Hajek, B. ``Gambler's Ruin: A Random Walk on the Simplex.'' §6.3 in Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, pp. 204-207, 1987.

Kraitchik, M. ``The Gambler's Ruin.'' §6.20 in Mathematical Recreations. New York: W. W. Norton, p. 140, 1942.