A bag holds 3 fair coins and 1 coin with a head on both sides. A coin is picked at random and tossed twice, and both tosses give a head. What is the probability that it is the two-headed coin?

Explanation

1. 01

   Two heads come from a fair coin with probability $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ and from the two-headed coin with probability 1, so $P(HH) = \frac{3}{4} \times \frac{1}{4} + \frac{1}{4} \times 1 = \frac{7}{16}$.

2. 02

   $\frac{1/4}{7/16} = \frac{4}{7}$.