WOLFRAM|DEMONSTRATIONS PROJECT

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.