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 limn→∞ α_n and limn→∞ P_n.