Question

A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events A: One of the balls is yellow B: At least one ball is red h C: Both balls are green D:Both balls are of the same color ) Find the following conditional probabilities: (a) P(BA)- b) P(DB)- (c) P(DIC)-

Answer 1

Total number of balls = 6

1 yellow, 2 red, 3 green

Bayes' Theorem: P(A | B) = P(A & B) / P(B)

Let Y indicate yellow, R indicate red and G indicate green

a) P(B | A) = P(one yellow and one red) / P(one of the balls is yellow)

= [P(YR) + P(RY)]/[P(YR) + P(RY) + P(YG) + P(GY)]

= (1/6 x 2/5 + 2/6 x 1/5) / (1/6 x 2/5 + 2/6 x 1/5 + 1/6 x 3/5 + 3/6 x 1/5)

= 0.4

b) P(D | B) = P(both are red)/P(at least one is red)

= P(both are red) / [1 - P(both are not red)]

= (2/6 x 1/5)/(1 - 4/6 x 3/5)

= 0.1111

c) P(D | C) = P(both are green)/P(both are green) = 1