Probability 2 Question 2
2. 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 equal to
(a) $\frac{1}{12}$
(b) $\frac{1}{6}$
(c) $\frac{1}{15}$
(d) $\frac{1}{9}$
(2002, 3M)
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Answer:
Correct Answer: 2. (c)
Solution:
- Given, $P(B)=\frac{3}{4}, P(A \cap B \cap \bar{C})=\frac{1}{3}$
and
$$ P(\bar{A} \cap B \cap \bar{C})=\frac{1}{3} $$
which can be shown in Venn diagram.
$\therefore P(B \cap C)=P(B)-{P(A \cap B \cap \bar{C}+P(\bar{A} \cap B \cap \bar{C}))}$
$$ =\frac{3}{4}-\frac{1}{3}+\frac{1}{3}=\frac{3}{4}-\frac{2}{3}=\frac{1}{12} $$
