Urn A has 17 white and 7 red balls Urn B has 11 white and 4

Urn A has 17 white and 7 red balls. Urn B has 11 white and 4 red balls. We flip a fair coin. If the oucome is heads, then a ball from urn A is selected, whereas if the oucome is tails, then a ball from urn B is selected. Suppose that a white ball is selected. What is the probability that the coin landed heads?

Solution

Let

A = urn A is selected (head)
B = urn B is selected (tails)
W = white ball is selected
R = red ball is selected

Thus,

P(A|W) = P(A and W) / P(W)

As

P(A and W) = P(A) P(W|A) = (1/2)(17/24) = 17/48

and

P(W) = P(A) P(W|A) + P(B) P(W|B) = (1/2)(17/24) + (1/2)(11/15) = 173/240

Then

P(A|W) = (17/48) / (173/240)

= 85/173 [answer]