Suppose there are two boxes B₁ and B₂ , each having 3 red and 4 black balls. One ball is drawn at random from B₁ . If it is red, 4 red balls are put into B₂ , otherwise 3 black balls are put into B₂ . Then one ball is randomly drawn from B₂ . If this ball is red, what is the conditional probability that the ball drawn from B₁ was also red?

Step-by-step solution

Let E₁ be the event that a red ball is drawn from B₁ . Let E₂ be the event that a black ball is drawn from B₁ . Let A be the event that a red ball is drawn from B₂ . The initial probabilities are: P(E₁) = 3 7 P(E₂) = 4 7 If E₁ occurs, 4 red balls are added to B₂ . The number of red balls in B₂ becomes 3 + 4 = 7 and black balls remain 4. Total balls = 11. P(A|E₁) = 7 11 If E₂ occurs, 3 black balls are added to B₂ . The number of red balls in B₂ remains 3 and black balls become 4 + 3 = 7 . Total balls = 10. P(A|E₂) =