"Consider two urns (A and B), each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are pA and pB, respectively. Balls are sequentially selected with replacement as follows. With probability P1, a ball is initially chosen from the first urn (A) and with probability 1 - P1, it is chosen from the second urn (B). 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 (and replaced), the next ball is taken from the other urn. We have to evaluate the expression -> Pn = αAB((βAB)^n−1 )+ γAB"
Solution: Let Pn denote the probability that the nth ball is chosen from the first urn (A). I got: Pn = (pA + pB − 1)Pn−1 +1 − pB Comparing with Pn = αAB((βAB)^n−1 )+ γAB, we get, αAB = P1 βAB = pA + pB − 1 and if we assume pA + pB − 1 = x and 1 − pB = y we get γAB = y * [ x^n-2 + x^n-3 + x^n-4 + . . . + x + 1 ]
Is this the correct way to approach the problem?