Question

An urn A contains 2 white & 4 black balls. Another urn B contains 5 white & 7 black balls. A ball is transferred from urn A to urn B, then a ball is drawn from urn B. Find the probability that it is white

Answer 1

Answer on Question #39526 – Math – Statistics and Probability

An urn A contains 2 white & 4 black balls. Another urn B contains 5 white & 7 black balls. A ball is transferred from urn A to urn B, then a ball is drawn from urn B. Find the probability that it is white.

Solution.

Let $A, B, C$ be three events:

$$A = \{ \text{the second ball is white} \};$$ $$B = \{ \text{the first ball is white} \};$$ $$C = \{ \text{the first ball is black} \};$$

We need to find $P(A)$. Hence:

$$P(B) = \frac{2}{6} = \frac{1}{3};$$ $$P(C) = \frac{4}{6} = \frac{2}{3};$$ $$P(A|B) = \frac{6}{13};$$ $$P(A|C) = \frac{5}{13};$$ $$P(A) = P(A \cap B) + P(A \cap C) = P(A|B)P(B) + P(A|C)P(C) = \frac{6}{13} \cdot \frac{1}{3} + \frac{5}{13} \cdot \frac{2}{3} = \frac{16}{39}.$$

Answer.

16/39