Question

Box I contains 7 red and 3 black balls; Box II contains 4 red and 5 black balls. After a randomly selected ball is transferred from Box I to Box II, 2 balls are drawn from Box II without replacement. Given that the two balls are red, what is the probability a black ball was transferred?

Answer 1

P(black ball transferred) = 3/10
P(Red ball transferred) = 7/10

When a ball is transferred from Box I to Box II there will be 10 balls in Box II and number of Red and Black will differ depending on the ball trasferred from Box I

required probability, P(black transferred | 2 Red) = P(2 Red | black transferred)*P(black transferred) / (P(2 Red | black transferred)*P(black transferred) + P(2 Red | red transferred)*P(red transferred))

P(2 Red | black transferred) = 4C2/10C2 = 0.1333
P(2 Red | red transferred) = 5C2/10C2 = 0.2222

P(black transferred | 2 Red) = 0.1333*0.3 / (0.1333*0.3 + 0.7*0.2222) = 0.2045

Ans: 0.2045