We wish to derive (3.22), the probability that B wins (i.e. that A goes bankrupt) in the gambler’s ruin problem.

(a) First assume p = 1/2. Let A be the event that person A is ruined, let W be the event that A wins the first round played, let q = 1 − p, and define si := Pri(A) := Pr(A | A starts with i dollars and B starts with T − i dollars).

Use the law of total probability to derive the difference equation

With r = q/p and di = si+1 − si, show that di = ri d0. Then determine the boundary conditions s0 and sT and use that s0 − sT = T −1 i=0 di to derive an expression for d0. Finally, write a similar telescoping expression for sj − sT , from which the answer follows.

(b) Derive the result for p = 1/2, either by similar steps as above, or with l’Hopital’s rule. ˆ

Si = psi+1+qsi-1, IiT