First slide

Theorems of probability

Question

$If A and B are two possible events which satisfy P (A \cup B) = P (A \cap B), then which of the following is always true?$

Moderate

A

$P (\frac{A}{B}) = P (A)$
B

$P (\frac{A}{B}) < 1 / 2$
C

$P (\frac{A}{B}) > 1 / 2$
D

$P (\frac{A}{B}) \neq P (\frac{B}{A})$

Solution

$\begin{array}{l} P (A \cup B) = P (A \cap B) \\ \Rightarrow A = B \\ \Rightarrow P (\frac{A}{B}) = \frac{P (A \cap B)}{P (B)} = \frac{P (B \cap B)}{P (B)} = \frac{P (B)}{P (B)} = 1 \end{array}$
$Similarly, P (\frac{B}{A}) = 1$

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