Question

Let $0 < P (A) < 1, 0 < P (B) < 1$ and $P (A \cup B) = P (A) + P (B) - P (A) P (B)$ then which of the following is correct?

Moderate

Solution

We know that
$\begin{aligned} P (A \cup B) = P (A) + P (B) - P (A \cap B) \\ ∴ Here, P (A \cap B) = P (A) P (B) \end{aligned}$
$\Rightarrow$ A, B are independent events.
$\Rightarrow A, \bar{B^{c}}; A^{c}, B and \bar{A^{c}}, \bar{B^{c}}$ also pairwise independent events.
$\begin{aligned} ∴ P (\bar{A \cup B}) = P (\bar{A} \cap \bar{B}) = P (\bar{A}) P (\bar{B}) \\ and P (A / B) = \frac{P (A \cap B)}{P (B)} = P (A) . \end{aligned}$

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