Bag I contains 3 white and 4 black balls, and Bag II contains 4 white and 5 black balls. One ball is transferred at random from Bag I to Bag II. Then a ball is drawn at random from Bag II. If the ball drawn from Bag II is found to be white, the probability that the transferred ball was black is m n , where m and n are coprime positive integers. The value of n - m is :

Step-by-step solution

Let E₁ be the event that a white ball is transferred from Bag I to Bag II, and E₂ be the event that a black ball is transferred. P(E₁) = 3 7 and P(E₂) = 4 7 . Let A be the event that the ball drawn from Bag II is white. If E₁ occurs, Bag II contains 5 white and 5 black balls. Thus, P(A|E₁) = 5 10 = 1 2 . If E₂ occurs, Bag II contains 4 white and 6 black balls. Thus, P(A|E₂) = 4 10 = 2 5 . By Bayes' Theorem, the probability that the transferred ball was black given that the drawn ball is white is: P(E₂|A) = P(E₂)P(A