I'm trying to work out the solution to a variant of the gambler's ruin. Here's my version:
There are two very unlucky but friendly gamblers A and B who decide to pool their money together to form a common budget with starting amount $b$, a positive integer. They roll a weighted die to decide who will play the next game. Therefore, gambler A will play a round with probability $p_A$. Likewise gambler B plays with probability $p_B=1-p_A$. Now, A and B are bad at gambling and either break even or lose money whenever they play—say \$1. So their pool of money can only decrease. However, they are not equally unlucky. Gambler A breaks even (does not lose or make money) $q_A$ of the time and loses otherwise. And gambler B breaks even $q_B$ of the time and loses otherwise.
When they've totally exhausted their funds, I want to know how much money each gambler is individually responsible for losing.
For example, say they started off with \$1000 and gambler A plays 1/3 of the time, and breaks even 1/3 of the times he plays. Gambler B plays (therefore) 2/3 of the time and breaks even 1/2 of the time she plays. (By simulation) gambler A is likely responsible for about \$400 lost and gambler B is responsible the remaining \$600.
I'd appreciate any hints.