If A, B and C are three events such that and , then P(B∩C) is equal to

Engineering

Mathematics

Conditional Probability

Question

If A, B and C are three events such that $P (B) = \frac{3}{4}, P (A \cap B \cap C^{'}) = \frac{1}{3}$ and $P (A^{'} \cap B \cap C^{'}) = \frac{1}{3}$ , then P(B∩C) is equal to

$\frac{1}{12}$

$\frac{1}{6}$

$\frac{1}{15}$

$\frac{1}{9}$

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We have

P(B∩C′) = P[(A∩A′)∩(B∩C′)]

P(B∩C′) = P(A∩B∩C′) + P(A′∩B∩C′)

$= \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$
Now, P(B∩C) = P(B) − P(B∩C′)

$= \frac{3}{4} - \frac{2}{3} = \frac{1}{12}$

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