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 $P (\bar{A}) = \frac{2}{3}$ then $P (\bar{A} \cap B)$ is-

$\frac{5}{12}$

$\frac{3}{8}$

$\frac{5}{8}$

$\frac{1}{4}$

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P(A) = 1 – P(A^')

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

P(B) = ?

$P (B) = \frac{2}{3}$

P(AUB) = P(A) + P(B) – P(A⋂B)

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

$P (B) = \frac{9 - 4 + 3}{12} = \frac{8^{2}}{123}$

$P (B) = \frac{2}{3}$

P(A'∩B) = P(B) – P(A∩B)

P(B) – P(A∩B) = P(AUB) – P(A)

$P (A^{'} \cap B) = \frac{3}{4} - \frac{1}{3} = \frac{9 - 4}{12} = \frac{5}{12}$

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