Bag A contains 3 white and 5 black balls, and Bag B contains 5 white and 3 black balls. One of the bags is chosen at random and 2 balls are drawn from it without replacement. Both drawn balls are found to be white. A third ball is then drawn from the same bag (without replacing the first two). If the probability that the third ball is white is p q , where p and q are coprime natural numbers, then the value of p + q i

Step-by-step solution

Let E₁ and E₂ be the events that Bag A and Bag B are chosen, respectively. P(E₁) = P(E₂) = 1 2 . Let W₂ be the event that the first two balls drawn are white. P(W₂|E₁) = ³C₂ ⁸C₂ = 3 28 P(W₂|E₂) = ⁵C₂ ⁸C₂ = 10 28 Using Bayes' Theorem, the posterior probabilities of the bags are: P(E₁|W₂) = P(E₁)P(W₂|E₁) P(E₁)P(W₂|E₁) + P(E₂)P(W₂|E₂) = 3 28 3 28 + 10 28 = 3 13 P(E₂|W₂) = 10 13 Now, we update the contents of the bags since 2 white balls have been removed: Updated Bag A contains 1 white and 5 black balls (total 6 ). Th