A bag contains 4 balls. Two balls are drawn at random & are found to be black. What is the probability that all the balls are black?
We need to compute P(all balls black | drew 2 black balls).
This is a conditional probability so we use Bayes Law.
P(A | B) = P(B | A) x P(A)/ P(B).
Where A = all balls black
B = draw 2 black balls
P(B) is not given. I will assume there are 3 possible worlds: 4 blacks, 3 blacks and 1 non black,
and 2 blacks and 2 non-black. 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| 4black) x P(4black) +P(B| 3black, 1non-black) x P(3black, 1 non-black) + P(B | 2black,2 non-black).
P(B|4black) = 1
P(4black) = 1/3 = P(3black and 2 non) = P(2black and 2 non)
P(B | 3black and 1 non) = 3/4 x 2/3 = 1/2
P(B | 2black and 1 non) = 2/4 x 1/3 = 1/6
Putting this all together with Bayes:
P(4black in bag | drew 2 black) = 1 x 1/3/ ( 1x1/3 + 1/2x1/3 + 1/6x1/3) = 3/5