Question

A bag contains $4$ balls. Two balls are taken out without replacement and found to be white. Find the probability that all the balls of the bag are white.

Answer 1

We need to compute P(all balls white | drew 2 white balls).

This is a conditional probability so we use Bayes Law.
P(A | B) = P(B | A) $\times$ P(A)/ P(B).
Where A = all balls white
B = draw 2 white balls
P(B) is not given. I will assume there are 3 possible worlds: 4 whites, 3 white and 1 non white,
and 2 whites and 2 non-white. I need to also know the probabilities for each of these worlds. I will assume they are equal, the standard assumption if no information is given.
P(B) = P(B| 4W)$\times$P(4W) +P(B| 3W, 1non-white)*P(3W, 1 non-white) + P(B | 2white,2 non-white).
P(B|4W) = 1
P(4W) = 1/3 = P(3W and 2 non) = P(2W and 2 non)
P(B | 3W and 1 non) = 3/4$\times$2/3 = 1/2
P(B | 2W and 1 non) = 2/4$\times$1/3 = 1/6
Putting this all together with Bayes:
P(4W in bag | drew 2 white) = 1$\times$ 1/3/ ( 1$\times$1/3 + 1/2$\times$1/3 + 1/6$\times$1/3) = 3/5