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 A bag contains 4 balls. Two balls are...
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Thu January 27, 2011 By: Aditya Sood
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?
Expert Reply
Dear Student,
Here is the solution:
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 nonblack. 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, 1nonblack) x P(3black, 1 nonblack) + P(B  2black,2 nonblack).
P(B4black) = 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
Regards
Team Topperlearning
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