In a classical ruin problem with two gamblers, the initial capital of one of the gamblers, Alice, is $90 and the combined initial capital (CIC) is $200. In each trial of the game, an unbiased coin is tossed - Alice wins when the coin shows heads.
a) What is the expected duration of the game?
b) Given the same CIC, how much more money should Alice have ititially so as to maximize the expected duration of the game?
Since the coin is unbiased, the probability of Alice winning (or losing) a dollar in a trial is 1/2. The numerical value of ...
This solution to this problem explains the probability of Alice winning a dollar per game. It also answers the question posted in detail.