Suppose there are two boxes B1 and B2 B1 contains two red ba

Suppose there are two boxes B1 and B2. B1 contains two red balls and one green ball. B2 contains one red ball and two green balls.   

a. A box is randomly selected, and then one ball is randomly selected from the box. The ball is red. What is the probability that the box selected is B1?

b. A box is randomly selected, and then two balls are randomly selected (without replacement) from the box. The first ball is red and the second ball is green. What is the probability that the box selected was B1?

Solution

a)

P(B1|red) = P(B1 and red) / P(red)

Here,

P(red) = P(B1) P(red|B1) + P(B2) P(red|B2) = (1/2)(2/3) + (1/2)(1/3) = 1/2

Also,

P(B1 and red) = P(B1) P(red|B1) = (1/2)(2/3) = 1/3

Thus,

P(B1|red) = (1/3) / (1/2) = 2/3 [answer]

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b)

P(red then green) = P(B1) P(red then green|B1) + P(B2) P(red then green|B2)

= (1/2)[(2/3)(1/2)] + (1/2)[(1/3)(1/2)]

P(red then green) = 1/4

Now,

P(B1|red then green) = P(B1) P(red then green|B1) / P(red then green)

= (1/2)[(2/3)(1/2)] / (1/4)

= 1/3 [answer]