Question 2 Consider the gambler's ruin with draws: Alice starts with fa and Bob with £(m - a), and at each time step Alice wins £1 off Bob with probability p, loses £1 to Bob with probability q, and no money is exchanged with probability s, where p +q + 8 =1. We consider the case where Bob and Alice are equally matched, so p= q and s = 1- 2p. (We assume 0<p <1/2.) Let ry be Alice's ruin probability from the point she has Li. (a) By conditioning on the first step, explain why pri+1-(1-s)r.+pr ;- 1 = 0, and give appropriate boundary conditions. [2] (b) Solve this linear difference equation to find an expression for ry- [2] Let d, be the expected duration of the game from the point Alice has &i. (c) Explain why pd141 - (1-8)d, + pd1-1 =- 1, and give appropriate boundary conditions. [2] (d) Solve this linear difference equation to find an expression for d .. [2] (e) Compare your answer to parts (b) and (d) with those for the standard gambler's ruin problem with p = 1/2, and give reasons for the similarities or differences. [2]

Fig: 1