*p*and tails with probability,

*q=1-p*(p can not = 1/2). If the coin lands heads, the gambler wins £1, whereas if the coin lands tails, the gambler loses £1.

Starting with £

*k*, the gambler plays the game repeatedly until he/she either goes bust or reaches £

*N*(

*N*>=

*k*). For

*k*=0,1,2,....

*N*let Ak be the event that the gambler starts with £

*k*and eventually goes bankrupt. Assume that the outcomes of the tosses are independent of each other.

(i) What are the values of P(A0) and P(AN)?

(ii) For

*k*=1,2,....

*N*-1, express

*P(Ak+1)*in terms of

*P(A*

*k*

*)*and

*P(Ak-1)*