The Gambler's Ruin

number of games in simulation

50

100

500

1000

5000

player's fixed win probability

0.5

player's initial stake

50

casino's initial stake

100

Probability of gambler's success: 0.333333

Expected number of plays until ruin or success: 5000

The gambler starts with an

i

unit stake and the casino or house starts with

c

units. They repeatedly play a game for which the gambler has a fixed probability

p

of winning and the winner gets 1 unit from the loser. Play continues until the gambler "succeeds" by acquiring

i+c

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

p=0.474

, the player's probability of winning an "even money" bet in American roulette.