32. In this exercise, we will outline a third technique for solving Example 3.31: We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement. What is the probability that an ace appears before a face card? Hint: Consider a Markov chain {Xn : n = 1, 2,...} with state space {1, 2, 3} and transition probability matrix P =   9/13 1/13 3/13 0 1 0 001   . The relation between the problem we want to solve and the Markov chain {Xn : n = 1, 2,...} is as follows: As long as a non-ace, non-face card is drawn, the Markov chain remains in state 1. If an ace is drawn before a face card, it enters the absorbing state 2 and will remain there indefinitely. Similarly, if a face card is drawn before an ace, the process enters the absorbing state 3 and will remain there forever. Let An be the event that the Markov chain moves from state 1 to state 2 in n steps. Show that A1 ⊆ A2 ⊆ ··· ⊆ An ⊆ An+1 ⊆ ··· , and calculate the desired probability P ! (∞ n=1 An " by applying Theorem 1.8: P *+∞ n=1 An , = lim n→∞ P (An) = lim n→∞ pn 12.