For k=1,2,3 the box B_k contains ' k ' red balls and (k+1) white balls. Let P (B₁ )= 1 2 , P (B₂ )= 1 3 and P (B₃ )= 1 6 . A box is selected at random, and a ball is drawn from it. If a red ball is drawn, then the probability that it has come from B₂ is

Step-by-step solution

Let R be the event that a red ball is drawn. The probabilities of selecting the boxes are P(B₁) = 1 2 , P(B₂) = 1 3 , and P(B₃) = 1 6 . The probabilities of drawing a red ball from each box are: P(R|B₁) = 1 1+2 = 1 3 P(R|B₂) = 2 2+3 = 2 5 P(R|B₃) = 3 3+4 = 3 7 Using the law of total probability, the probability of drawing a red ball is: P(R) = P(B₁)P(R|B₁) + P(B₂)P(R|B₂) + P(B₃)P(R|B₃) P(R) = ( 1 2 1 3 ) + ( 1 3 2 5 ) + ( 1 6 3 7 ) P(R) = 1 6 + 2 15 + 1 14 Finding a common denominator, which is 210 : P(R) = 35 210