The probability that at least one of the events A and B occurs is . If A and B occur simultaneously with probability then P(A′)+P(B′) is

Engineering

Mathematics

Conditional Probability

Question

The probability that at least one of the events A and B occurs is $\frac{3}{5}$ . If A and B occur simultaneously with probability $\frac{1}{5}$ then P(A′)+P(B′) is

$\frac{2}{5}$

$\frac{4}{5}$

$\frac{6}{5}$

$\frac{7}{5}$

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$P (A \cup B) = \frac{3}{5}$ and $P (A \cap B) = \frac{1}{5}$

P(A∪B) = P(A) + P(B) − P(A∩B)

⇒ P(A) + P(B) = P(A∪B) + P(A∩B)

$= \frac{3}{5} + \frac{1}{5} = \frac{4}{5}$

$\Rightarrow 1 - P (A) + 1 - P (B) = \frac{4}{5}$

$\Rightarrow P (A) + P (B) = 2 - \frac{4}{5} = \frac{6}{5}$

$\Rightarrow P (A^{c}) + P (B^{c}) = \frac{6}{5}$

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