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...

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 {X_n
: n = 1, 2,...} with state space {1, 2, 3} and transition probability matrix

The relation between the problem we want to solve and the Markov chain {X_n
: 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

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?

Theorem 1.8

(Continuity of Probability Function)

For any increasing or decreasing sequence of events, {E_n, n ≥ 1},