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.

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

Let ${E_1} =$ Event that bag ${\rm{I}}$

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

is selected
and $E =$ Event that a black ball is selected

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

$\therefore$ $P(E) = P\left( {{E_1}} \right) \cdot P\left( {E/{E_1}} \right) + P\left( {{E_2}} \right) \cdot P\left( {E/{E_2}} \right)$

$= \frac{1}{2} \cdot \frac{3}{5} + \frac{1}{2} \cdot \frac{2}{6} = \frac{3}{{10}} + \frac{2}{{12}}$
$= \frac{{18 + 10}}{{60}} = \frac{{28}}{{60}} = \frac{7}{{15}}$