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Each morning a student takes one of the three books he owns from his shelf. The probability that he chooses book $i$ is $a_i$, where $0 < a_i < 1$ for $i=1,2,3$ and the choices on successive days are independent.

In the evening he replaces the book at the left-hand end of the shelf. If $P_n$ denotes the probability that on day $n$ the student finds the books in the order $1,2,3$, from left to right, show that, irrespective of the initial arrangement of the books, $P_n$ converges as $n\to\infty$ and find the limit.

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    is the solution for all six probabilities ie: a1a2/a2+a3 because then the probabilities don't actually sum to 1. shouldn't it be p123=a1a2/a2+a3, p213=a2a1/a1+a3,etc2012-09-04

4 Answers 4