Suppose there are 2 banks with equal money units. A fair coin toss decides which bank loses 1 unit of money to the tosser i.e. if result is heads, Bank A loses 1 unit and if the result is tails, Bank B loses 1 unit. Bank A always wins with heads and bank B always wins with tails. The tosser is a third party in the game. The game stops when a bank runs out of money and then loses the toss.
What is the expected payout of the tosser?
This is called Banach's matchbox puzzle and a full solution (for number of money units left in the non-losing bank) is given. From that you can calculate its expectation and then the tosser's expected profit.
If you're looking for a hint (before checking wikipedia), the solution involves the negative binomial distribution.