Three coins

A bag holds a fair coin, a two-headed coin and a two-tailed coin. You draw one at random, flip it, and see heads. What is the probability it is the two-headed coin?

Hint

Use Bayes’ rule. First find the overall probability of seeing heads.

Solution

$P(H) = \tfrac13\left(\tfrac12 + 1 + 0\right) = \tfrac12$

$P(\text{two-headed}\mid H) = \frac{\tfrac13 \cdot 1}{\tfrac12} = \tfrac23$

Answer

$2/3$