Bag I has 3 red and 2 blue balls; bag II has 3 red and 1 blue ball. A bag is picked at random and two balls are drawn from it, one after the other, without replacement. Both are red. What is the probability that bag II was picked?

Explanation

1. 01

   $P(RR \mid \text{I}) = \frac{3}{5} \times \frac{2}{4} = \frac{3}{10}$ and $P(RR \mid \text{II}) = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$, so $P(RR) = \frac{1}{2} \times \frac{3}{10} + \frac{1}{2} \times \frac{1}{2} = \frac{2}{5}$.

2. 02

   $P(\text{II} \mid RR) = \frac{\frac{1}{2} \times \frac{1}{2}}{\frac{2}{5}} = \frac{5}{8}$.