Two gamblers, call them Gambler A and Gambler B flip a coin repeatedly. The coin is unfair and comes up heads 2/3 of the time. Gambler A wins one dollar from Gambler B, when a head is tossed. Conversely Gambler B wins one dollar from Gambler A when a tail is tossed. The coin tosses are independent. The game ends when one of the gamblers runs out of money. There are 5 dollars in the pot. Determine the probability that Gambler A wins the game if he starts with I dollars.Here I = 1, 2, 3, or 4. 1 0 or 5 are absorbing states, so that they cannot be used as initial states. This means that if Gambler A has zero dollars, he is ruined and if Gambler A has five dollars, he won the game.

Fig: 1