Bag I contains 3 black and 2 white balls, bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

$\frac{7}{15}$

The correct answer is Option (2) → $\frac{7}{15}$ ##

Bag I $= \{3B, 2W\}$, Bag II $= \{2B, 4W\}$

Let $\quad E_1 = \text{Event that bag I is selected}$

$E_2 = \text{Event that bag II is selected}$

and $E = \text{Event that a black ball is selected}$

$\Rightarrow P(E_1) = 1/2, P(E_2) = \frac{1}{2}, P(E | E_1) = \frac{3}{5}, P(E | E_2) = \frac{2}{6} = \frac{1}{3}$

$∴P(E) = P(E_1) \cdot P(E | E_1) + P(E_2) \cdot P(E | E_2)$

$= \frac{1}{2} \cdot \frac{3}{5} + \frac{1}{2} \cdot \frac{1}{3} = \frac{3}{10} + \frac{1}{6} = \frac{18 + 10}{60} = \frac{28}{60} = \frac{7}{15}$