First slide

Theorems of probability

Question

$It is given that the events A and B are such that P (A) = \frac{1}{4}, P (\frac{A}{B}) = \frac{1}{2}, P (\frac{B}{A}) = \frac{2}{3} then P (B) =$

Moderate

A

$\frac{1}{6}$
B

$\frac{1}{3}$
C

$\frac{2}{3}$
D

$\frac{1}{2}$

Solution

$\begin{array}{l} P (\frac{B}{A}) = \frac{P (A \cap B)}{P (A)} \\ \Rightarrow \frac{2}{3} = \frac{P (A \cap B)}{1 / 4} \\ \Rightarrow P (A \cap B) = \frac{1}{4} \cdot \frac{2}{3} = \frac{1}{6} \\ P (\frac{A}{B}) = \frac{P (A \cap B)}{P (B)} \Rightarrow \frac{1}{2} = \frac{1 / 6}{P (B)} \\ \Rightarrow P (B) = \frac{1 / 6}{1 / 2} = \frac{2}{6} = \frac{1}{3} \end{array}$

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