First slide

Theorems of probability

Question

If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$ , then $P (B \cap C)$ is

Difficult

A

$\frac{1}{12}$
B

$\frac{1}{6}$
C

$\frac{1}{15}$
D

$\frac{1}{9}$

Solution

Given that $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$
$P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$ .
From Venn's diagram, we have
$\begin{array}{l} P (B \cap C) = P (B) - P (A \cap B \cap \bar{C}) - P (\bar{A} \cap B \cap \bar{C}) \\ = \frac{3}{4} - \frac{1}{3} - \frac{1}{3} \\ = \frac{9 - 4 - 4}{12} \\ = \frac{1}{12} \end{array}$

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