I have 10 Urns; 9 contain 2 White and 2 Black Balls, one urn contains 5 White and one Black ball. A ball chosen from a random urn is white. What is the probability that it came from the urn with the five white balls?
I know I have to use Bayes Theorem here, but I am not quite sure how to use it in this given case. Any help is highly appreciated.
Thanks, Daniel
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To elaborate a little on the previous answer. Number your urns 1 through 10, with the tenth urn being the one with 5 white and 1 black ball, and let P(10|W) denote the conditional probability that the ball came from the 10th urn, given that it is white.
From the definition of conditional probability, you know:
P(10|W) = P(W and 10th)/P(W)
You can compute the probability that it came from the tenth urn and is white, since this is just (1/10)(5/6). So the question is what is P(W)? But we can decompose this event as the sum of the disjoint events:
P(W) = P(W and 1) + P(W and 2) + .... P(W and 10)
Note that if n is not equal to 10 (i.e. n=1,2,...,9), then P(n and W) = (1/10)*(1/2), so
P(W) = (1/10)(9/2 + 5/6) = 32/60 = 8/15
Bayes theorem says that: $$ P(A|B)P(B)=P(B|A)P(A) $$ let $A$ be the event of the urn you draw from having 5 white balls, and $B$ be the event 'draw a white ball'. Now, you want $P(A|B)$, and this can be expressed in the other three probabilities, which are easier to calculate.