Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) sel...

Question

Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) select a box, (ii) choose a ball from the selected box such that each ball in the box is equally likely to be chosen. The probabilities of selecting boxes P and Q are 1/3 and 2/3 respectively. Given that a ball selected in the above process is a red ball, the probability that it came from box P is

A. 4/19
B. 5/19
C. 2/9
D. 19/30

Accepted Answer

P(box P | red ball) = 4/19. Answer: A. This is a classic Bayes' theorem problem. A box is chosen first (P with probability 1/3, Q with probability 2/3), then a ball is drawn uniformly from that box. We observe that the drawn ball is red, and we want the posterior probability that it came from box P.