Consider two urns, each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are, respectively, p and p' Balls are sequentially selected with replacement as follows: With probability α, a ball is initially chosen from the first urn, and with probability 1 - α, it is chosen from the second urn. The subsequent selections are then made according to the rule that whenever a white ball is drawn (and replaced), the next ball is drawn from the same urn, but when a black ball is drawn, the next ball is taken from the other urn. Let α_n denote the probability that the nth ball is chosen from the first urn. Show that

and use this formula to prove that

Let P_n denote the probability that the nth ball selected is white. Find P_n. Also, compute lim_n→∞an and lim_n→∞Pn.

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An+1 = an (p + p' – 1) +1- p' n2 1