The Gambler's Ruin
The gambler starts with an unit stake and the casino or house starts with units. They repeatedly play a game for which the gambler has a fixed probability of winning and the winner gets 1 unit from the loser. Play continues until the gambler "succeeds" by acquiring units or is "ruined" by dropping to 0 units. This Demonstration computes the probability that the gambler will succeed by breaking the bank. Subtracting this probability from 1 gives the gambler's ruin probability. The theoretical expected number of plays of the game until success or ruin is also computed and a simulation gives empirical results for the various parameter values. In the example shown in the thumbnail we use , the player's probability of winning an "even money" bet in American roulette.
Snapshot 1: a fair game—in such cases the gambler's overall success probability is simply , the gambler's proportion of the total stake and the expected number of plays until ruin is
Snapshot 2: , which is about what an expert card counter in blackjack might achieve; in the simulation, the gambler generates a nice profit, but would require a very long time to break the bank
Snapshot 3: results also apply to two individuals playing "head on"; here the "player" triumphs with the help of a larger bankroll, despite the fact that the odds slightly favor the opponent
Related results: Derivation of the relevant formulas for probability of success and for expected time for success or ruin involves a nice application of recurrence relations. It also provides an interesting example of an absorbing Markov chain. For a more elementary approach, see Chapter 6 of the reference below.
E. Packel, The Mathematics of Games and Gambling, 2nd ed., Washington: The Mathematical Association of America, 2006.