Question

Bag A contains 2 White and 4 Black balls. Another bag B contains 5 white and 7 black balls. A ball is transformed from the bag A to the bag B. Then a ball is drawn from the bag B. Find the probability that it is Black.

Answer 1

Answer on Question #46077 – Math – Statistics and Probability

Problem.

Bag A contains 2 White and 4 Black balls. Another bag B contains 5 white and 7 black balls. A ball is transformed from the bag A to the bag B. Then a ball is drawn from the bag B. Find the probability that it is Black.

Solution:

Let $X, Y$ are events: $X =$ "the black ball was taken from the bag $A$", $Y =$ " the black ball will be taken from the bag $B$". Hence

$$P(X) = \frac{4}{2 + 4} = \frac{2}{3} \text{ and } P(X^c) = 1 - P(X) = 1 - \frac{2}{3} = \frac{1}{3},$$ $$P(Y|X) = \frac{7 + 1}{5 + (7 + 1)} = \frac{8}{13} \text{ and } P(Y|X^c) = \frac{5 + 1}{(5 + 1) + 7} = \frac{6}{13}.$$

Therefore

$$P(Y) = P(Y|X)P(X) + P(Y|X^c)P(X^c) = \frac{2}{3} \cdot \frac{8}{13} + \frac{1}{3} \cdot \frac{6}{13} = \frac{16 + 6}{39} = \frac{22}{39}$$

by the Law of total probability.

Answer: $P(Y) = \frac{22}{39}$.

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