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

100

200

300

400

500

number of bets

10

20

30

40

50

60

current stake of the gambler

Neither the gambler nor the house was ruined.

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.