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# 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.

Toolbox:
• If A and B are independant events, $P(A\cap\;B)=P(A)\;P(B)$
Step 1:
Let $E_1$ be the event of choosing the bag I
Let $E_2$ be the event of choosing the bag II
Let A be the event of drawing a black ball
$\therefore P(E_1)=P(E_2)=\large\frac{1}{2}$
Step 2:
Now $P(\large\frac{A}{E_1})$=P(drawing a black ball from bag I)
$\Rightarrow \large\frac{3}{5}$
$P(\large\frac{A}{E_2})$=P(drawing a black ball from bag II)
$\Rightarrow \large\frac{2}{6}$
$\Rightarrow \large\frac{1}{3}$
Step 3:
$\therefore$ Probability of selecting a black ball is $\large\frac{1}{2}\times \frac{3}{5}+\frac{1}{2}\times \frac{1}{3}$
$\Rightarrow \large\frac{3}{10}+\frac{1}{6}$
$\Rightarrow \large\frac{7}{15}$