involving the selection of a colored ball from one of three urns. Assume that yo
ID: 2956764 • Letter: I
Question
involving the selection of a colored ball from one of three urns. Assume that you randomly pick one of the urns: X, Y, or Z.You then randomly draw one ball out of the selected urn and note its color. The urns contain the following colored balls:Urn X: 2 red, 2 white, and 1 blue.
Urn Y: 1 red, 2 white, and 1 blue.
Urn Z: 2 red, 2 white, and 3 blue.
(1) If the ball is blue, what is the probability that it was drawn from urn X?
(2) If the ball is either blue or white, what is the probability that it was drawn
from urn Y?
Explanation / Answer
(1) P (X |B) = P (X n B) / P(B) P (X n B) = P (B | X) P(X) = (2/8) (1/3) P(B) = P(B|X) P(X) + P(B|Y) P(Y) + P(B|Z) P(Z) = (2/8)(1/3) + (3/7)(1/3) + (1/5)(1/3) so P (X |B) = (2/8) / ( (2/8) + (3/7) + (1/5)) = 35/123 = 0.284553 (2) P(Y | B+W) = P ( Y n (B+W)) / P(B+W) P ( Y n (B+W)) = P ( B+W | Y) P(Y) = (5/7) (1/3) P (B+W) = P(B+W|X) P(X) + P(B+W|Y) P(Y) + P(B+W|Z) P(Z) = (7/8)(1/3) + (5/7)(1/3) + (2/5)(1/3) so P(Y | B+W) = (5/7) / ( (7/8) + (5/7) + (2/5) ) = 200/557 = 0.359066
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