I cannot solve this problem. Does anyone know how I can solve it? I already have my state space and transition matrix.
Problem: Each morning, a student takes one of the three books (labelled 1,2 and 3) from her shelf. She choose book i with probability alpha_i, and choices on successive days are independent. In the evening, she replaces the book at the left-hand end of the shelf. If p_n denotes the probability that on day n she find the books in the order 1,2,3 from left to right, show that p_n converges as n goes to infinity, and find the limit.
hint: $X_t=$the order of the books in the morning
so, $\Pr(X_{t+1}=1,2,3|X_t=3,2,1)=0$, $\Pr(X_{t+1}=1,2,3|X_t=2,1,3)=\alpha_1$, and so on