If A and B are events such that and , then is

Engineering

Mathematics

Conditional Probability

Question

If A and B are events such that $P (A \cup B) = \frac{3}{4}, P (A \cap B) = \frac{1}{4}$ and $P (A) = \frac{1}{3}$ , then $P (A \cap B)$ is

$\frac{5}{12}$

$\frac{3}{8}$

$\frac{5}{8}$

$\frac{1}{4}$

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$P (A \cup B) = \frac{3}{4} \Rightarrow P (\bar{A \cap B}) = \frac{3}{4} \Rightarrow P (A \cap B) = \frac{1}{4}$

$P (A \cap B) = \frac{1}{4} \Rightarrow P (\bar{A \cup B}) = \frac{1}{4} \Rightarrow P (A \cup B) = \frac{3}{4}$

$P (A) = \frac{1}{3}, P (A) = \frac{2}{3}$

$P (A \cap B) = P (B) - P (A \cap B)$

P(A∪B) = P(A) + P(B) − P(A∩B)

$\Rightarrow \frac{3}{4} = \frac{1}{3} + P (B) - \frac{1}{4}$

$P (B) = \frac{3}{4} - \frac{1}{12} = \frac{9 - 1}{12} = \frac{8}{12} = \frac{2}{3}$

$∴ P (A \cap B) = \frac{2}{3} - \frac{1}{4} = \frac{8 - 3}{12} = \frac{5}{12}$

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