Bag I has 3 red balls and 1 black ball; bag II has 2 red and 2 black balls. A bag is picked at random and a ball is drawn from it; $R$ is the event that the ball is red. Which of the following statements are true?

I. $P(\text{I} \mid R) = \frac{3}{5}$.

II. $P(\text{I} \mid R) + P(\text{II} \mid R) = 1$.

III. $P(\text{I} \mid R) + P(\text{I} \mid R') = 1$.

IV. $P(\text{II} \mid R') = \frac{2}{3}$.

Explanation

1. 01

   $P(R) = \frac{1}{2} \times \frac{3}{4} + \frac{1}{2} \times \frac{2}{4} = \frac{5}{8}$, so $P(\text{I} \mid R) = \frac{3/8}{5/8} = \frac{3}{5}$; $P(R') = \frac{3}{8}$, so $P(\text{II} \mid R') = \frac{2/8}{3/8} = \frac{2}{3}$. I and IV are true.

2. 02

   II is true: given $R$, the ball came from exactly one bag. III is false: $P(\text{I} \mid R') = \frac{1}{3}$, and $\frac{3}{5} + \frac{1}{3} = \frac{14}{15}$.