First slide

Theorems of probability

Question

$Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A \cup B) = P (A) + P (B) - P (A) P (B), then$

Moderate

A

$P (A ∣ B) = 0$
B

$P (B ∣ A) = 0$
C

$P (A^{'} \cap B^{'}) = P (A^{'}) P (B^{'})$
D

$P (A ∣ B) + P (B ∣ A) = 1$

Solution

We know that
$\begin{array}{l} P (A \cup B) = P (A) + P (B) - P (A \cap B) \\ \Rightarrow P (A) + P (B) - P (A) P (B) = P (A) + P (B) - P (A \cap B) \\ \Rightarrow P (A) \cdot P (B) = P (A \cap B) \\ \Rightarrow A and B are independent. \end{array}$

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