Question

If A and B are two independent events such that P(A) = 1/2 and P(B) = 1/5, then

Moderate

Solution

Since A and B are independent events, we have
$\begin{array}{l} P (A \cap B) = P (A) P (B) = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \\ P (A / B) = P (A) = \frac{1}{2} \end{array}$
Now,
$\begin{array}{l} P (A \cup B) = P (A) + P (B) - P (A \cap B) \\ = \frac{1}{2} + \frac{1}{5} - \frac{1}{10} = \frac{3}{5} \end{array}$
Now,
$\begin{array}{l} P (A / A \cup B) = \frac{P [A \cap (A \cup B)]}{P (A \cup B)} = \frac{P (A)}{P (A \cup B)} = \frac{1 / 2}{3 / 5} = \frac{5}{6} \\ P [(A \cap B) / (A \cup B)] = P (A \cap B / (A \cap B)) = 0 \end{array}$

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