For k =1,2,3 the box B _ k contains k red balls and ( k +1) white balls. Let P ( B ₁ )= 1 2 , P ( B ₂ )= 1 3 and P ( B ₃ )= 1 6 . A box is selected at random and a ball is drawn from it. If a red ball is drawn from it, then the probability

Question

For k =1,2,3 the box B _ k contains k red balls and ( k +1) white balls. Let P ( B ₁ )= 1 2 , P ( B ₂ )= 1 3 and P ( B ₃ )= 1 6 . A box is selected at random and a ball is drawn from it. If a red ball is drawn from it, then the probability that it comes from box B₂ is

Quantrex Academy · Accepted Answer

Correct answer: B. 14 39

Let B_k be the event that box k is selected ( k=1,2,3 ) and R be the event that a red ball is drawn. The prior probabilities are P(B₁) = 1 2 , P(B₂) = 1 3 , and P(B₃) = 1 6 . Box k contains k red balls and (k+1) white balls, totaling 2k+1 balls. The conditional probabilities are therefore P(R|B₁) = 1 3 , P(R|B₂) = 2 5 , and P(R|B₃) = 3 7 . The total probability of drawing a red ball is found using the law of total probability: P(R) = 1 3 1 2 + 2 5 1 3 + 3 7 1 6 = 1 6 + 2 15 + 1 14 . Using a common denominator of 21

For k =1,2,3 the box B _ k contains k red balls and ( k +1) white balls. Let P ( B ₁ )= 1 2 , P ( B ₂ )= 1 3 and P ( B ₃ )= 1 6 . A box is selected at random and a ball is drawn from it. If a red ball is drawn from it, then the probability that it comes from box B₂ is

Options

Correct answer

Step-by-step solution

More from Probability