Introduction to Probability · Problem 12.2.1

Random Walks — Grinstead & Snell

Mathematics — Probability · gamblers-ruin-generating-functions

Problem

In the gambler's ruin problem, assume that the gambler initial stake is 1 dollar, and assume that her probability of success on any one game is

p

. Let

T

be the number of games until 0 is reached (the gambler is ruined). Show that the generating function for

T

h(z) = \frac{1 - \sqrt{1 - 4pqz^2}}{2pz},

and that

h(1) = \begin{cases} q/p, & \text{if } q \leq p, \\ 1, & \text{if } q \geq p, \end{cases}

and

h'(1) = \begin{cases} 1/(q-p), & \text{if } q > p, \\ \infty, & \text{if } q = p. \end{cases}

Interpret your results in terms of the time

T

to reach 0. (See also Example 10.7.)