Question

I'm stuck on this Probability?

Two players A and B play the following game: Initially, player A has n(sub)A pounds and player
B has n(sub)B pounds, both greater than zero. They toss a fair coin and if it turns HEADS, then player B gives one pound to
A. If it turns TAILS, then player A gives a pound to B. The game stops when one of the players loses
all his money. What is the average number of steps until the end of the game ?

(HINT: let m(sub)j be the expected number of steps required, when player A has j pounds and try to set up a recursive equation
for m(sub)j.)

Answer 1

if we start with each player having A and B, the expected value is A*B

let (a,b) be the expected number of steps, given player A has a, and B has b

I started with some simple combinations (1,1) is 1 step
(1,2) is 1/2 (1 + (0,3)) + 1/2 (1 + (2,1))= 1 + 1/2(1,2) (by symmetry (1,2)= (2,1)
or 1/2 (1,2)= 1 and (1,2)= 2
Continuing in that way, I found (1,3)= (3,1)=3 (2,2)= 4 and so on.
At this point I think we need an inductive proof to show this rule holds in general.