Q.Bag I contains 4 red balls and 7 white balls. Bag II contains 5 red balls and 6 white balls. One of the bags is chosen at random and a ball is drawn from it. Find the probability of drawing:

Use the law of total probability: average the conditional probability of each outcome over the two equally-likely bag choices.

Let $E_1$ = choosing Bag I, $E_2$ = choosing Bag II, each with probability $\frac12$.

Bag I: 4 red, 7 white (11 total). Bag II: 5 red, 6 white (11 total).

1. Probability of drawing a red ball: $P(\text{red}) = P(E_1)P(\text{red}|E_1) + P(E_2)P(\text{red}|E_2) = \frac12\cdot\frac{4}{11} + \frac12\cdot\frac{5}{11} = \frac12\left(\frac{9}{11}\right) = \frac{9}{22}$
2. Probability of drawing a white ball: …