
Gambler's Ruin
A gambler bets a stake of £1 repeatedly on a game. They lose with probability 1/2 in which case the stake is lost. If the gambler wins they are, with equal probability, either returned their £1 stake or they recieve their £1 and an additional £1.
Starting with £k the gambler playes the game repeatedly until they either go bankrupt or reach £N, where N greater or equal to k.
Let $\displaystyle B_{k}$ be the event that the gambler eventually goes bankrupt.
For k=1,2,...N1 show that:
$\displaystyle 3P(B_{k})=P(B_{k+1})+2P(B_{k1})$
I know the answer to this should be pretty simple, but I can't get it to work. A thorough solution would be of much help! Thanks