Ben and Jerry play a series of games of cards. During each game each player bets a $1, and whoever wins the game gets the $2. Sometimes a game can end in a tie, in which case neither player loses his money. Ben is a better player than Jerry and has a probability 0.5 of winning each game, a probability 0.3 of losing each game, and probability 0.2 of tying with Jerry. Initially Ben had $9, while Jerry had $6, and the game is over when either player is wiped out.
a. Give the state transition diagram of the process.
b. If rk denotes the probability that Ben is ruined, given that the process is currently in state k, obtain an expression for rk in the first game when the process is in state k.