A woman needs \($5\),000 for a down payment on a condominium. She will try to raise the money for the down payment by gambling. She will place a sequence of bets until she either accumulates \($5\),000 or loses all her money. She starts with \($2\),000, and will wager \($1\),000 on each bet. Each time that she bets \($1\),000, she will win \($1\),000 with probability 0.4, or lose \($1\),000 with probability 0.6.

(a) Model the woman’s gambling experience as a six-state absorbing multichain. Let the state X_n denote the gambler’s revenue after the nth bet. Construct the transition probability matrix.

(b) What is the expected number of bets that she will make?

(c) What is the probability that she will obtain her \($5\),000 down payment?