Suppose you are at a casino and you have 1 dollar. You have found a game that costs one dollar to play and you will win 2 dollars (your original bet plus one extra dollar) with 2/3 probability and lose with 1/3 probability. Once you have no money left you will go home.

Suppose you are at a casino and you have 1 dollar. You have found a game that costs one dollar to play and you will win 2 dollars (your original bet plus one extra dollar) with 2/3 probability and lose with 1/3 probability. Once you have no money left you will go home. (a) (5 points) Let Pn be the probability that you go home after n games. Compute Pi, P2, Ps, P, Ps, Pe, Pr and explain your answers. (you may use pictures to explain.) (b) Let C1 be the probability that you eventually run out of money if you start with 1 dollar and C2 be the probability that you eventually run out of money if you start with 2 dollars. i. (2 points) Explain why C,- + . ii. (2 points) Explain why = C? ii. (3 points) Using the two equations above, solve for C1 (Assume that 1. This can actually be shown using an argument involving differential equations. But I am asking that you just take it to be true without proof.)