A fair die is tossed repeatedly. We begin studying the outcomes after the first 6 occurs. Let the first 6 be called the zeroth outcome, let the first outcome after the first six, whatever it is, be...

A fair die is tossed repeatedly. We begin studying the outcomes after the first 6 occurs. Let the first 6 be called the zeroth outcome, let the first outcome after the first six, whatever it is, be called the first outcome, and so forth. For n ≥ 1, define X_n
= i if the last 6 before the nth outcome occurred i tosses ago. Thus, for example, if the first 7 outcomes after the first 6 are 1, 4, 6, 5, 3, 1, and 6, then X₁
= 1, X₂
= 2, X₃
= 0, X₄
= 1, X₅
= 2, X₆
= 3, and X₇
= 0. Show that {X_n
: n = 1, 2,...} is a Markov chain, and find its transition probability matrix. Furthermore, show that {X_n
: n = 1, 2,...} is an irreducible, positive recurrent, aperiodic Markov chain. For i ≥ 1, find πi, the long-run probability that the last 6 occurred i tosses ago.