Question
Sun February 06, 2011 By:

A BAG CONTAINS 4 BALLS. TWO BALLS ARE DRAWN AT RANDOM, AND ARE FOUND TO BE WHITE. WHAT IS THE PROBABILITY THAT ALL BALLS ARE WHITE?

Expert Reply
Mon February 07, 2011
Dear Student,
Here is the solution:

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) * 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)*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*2/3 = 1/2

P(B | 2W and 1 non) = 2/4*1/3 = 1/6

Putting this all together with Bayes:
P(4W in bag | drew 2 white) = 1* 1/3/ ( 1*1/3 + 1/2*1/3 + 1/6*1/3) = 3/5
Regards
Team Topperlearning
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