Question

Urn $1$ contains $5$ white balls and $7$ black balls. Urn $2$ contains $3$ white and $12$ black. A fair coin is flipped, if it is heads, a ball is drawn from urn $1$, and if it is tails, a ball is drawn from urn $2$. Suppose that this experiment is done and a white ball was selected. What is the probability that this ball was in fact taken from urn $2$? (i.e., that the coin flip was tails)

• A. $\frac{15}{37}$
• B. $\frac{3}{10}$
• C. $\frac{12}{37}$
• D. $\frac{1}{24}$

Answer 1

Answer: (C)

Let $T$ be the event that the coin flip was tails. Let $W$ be the event that a white ball is select. From the given data, we know that $P(W|T) = 3/15$ and that $P(W|T^c) = 5/12$. Since the coin is fair, we know that $P(T) = P(T^c) = 1/2$.
Required probability :
$P\left(T\, |\,W\right)= \frac{P\left(T\cap W\right)}{P\left(W\right)}$

$= \frac{P\left(W \,|\,T\right)\,PT}{P\left(W\,|\, T\right) P\left(T\right) + P\left(W\,|\, T^{c}\right)P\left(T^{c}\right)}$

$= \frac{\left(3/15\right)\left(1/2\right)}{\left(3/15\right)\left(1/2\right) +\left(5/12\right)\left(1/2\right)} =\frac{12}{37}$