Grinstead-Snell-Probability · Problem 11.2.23

Markov Chains

Mathematics · absorbing-markov-chains

Problem

A gambler plays a game in which on each play he wins one dollar with probability

p

and loses one dollar with probability

q = 1 - p

. The Gambler's Ruin problem is the problem of finding the probability

w_x

of winning an amount

T

before losing everything, starting with state

x

. Show that this problem may be considered to be an absorbing Markov chain with states

0, 1, 2, \ldots, T

with 0 and

T

absorbing states. Suppose that a gambler has probability

p = .48

of winning on each play. Suppose, in addition, that the gambler starts with 50 dollars and that

T = 100

dollars. Simulate this game 100 times and see how often the gambler is ruined. This estimates

w_{50}