Suppose that the woman seeking a \($5\),000 down payment in Problem 3.3 has the option of betting either \($1\),000 or \($2\),000. She chooses the following aggressive strategy. If she has \($1\),000 or \($4\),000, she will bet \($1\),000, and will win \($1\),000 with probability 0.4 or lose \($1\),000 with probability 0.6. If she has \($2\),000 or \($3\),000, she will bet \($2\),000, and will either win \($2\),000 with probability 0.05, or win \($1\),000 with probability 0.15, or lose \($2\),000 with probability 0.8. 

Data in Problem 3.3

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 under the aggressive strategy as a six-state absorbing Markov chain. Let the state represent the amount of money that she has when she places a 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?