Player $A$ and player $B$ are playing coin tossing game. Player $A$ has $n$ coins, player B has $m$ coins. If two head or tails turn up then player $A$ takes both coins. Otherwise player $B$ takes them.
I'm thinking this way:
Let us define $F_{n,m}$ - expected game length if $A$ has $n$ coins and $B$ has $m$. Probability of winning in a single round for both players is equal.
Then one has $$F_{n,m} = 1 + 1/2F_{n-1,m} + 1/2 F_{n,m-1}$$
Obviously $F_{0, m} = 0$ and $F_{n,0} = 0$.
How do I solve this reccurence relation? Or should I went differently about this problem?