Consider two urns, each containing both white and black

Chapter 3, Problem 20TE

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Problem 20TE

Consider two urns, each containing both white and black balls. The probabilities of drawing white bails from the first and second urns are, respectively, p and p′. Balls are sequentially selected with replacement as follows: With probability a, 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 αn denote the probability that the nth ball is chosen from the first urn. Show that

and use this formula to prove that

Let Pndenote the probability that the nth ball selected is white. Find Pn.Also, compute limn → ∞αn and limn → ∞ Pn.