If for two events A and B,and then equals

Engineering

Mathematics

Conditional Probability

Question

If for two events A and B, $P (A \cup B) = \frac{1}{2}, P (A \cap B) = \frac{2}{5}$ and then $P (A^{c}) + P (B^{c})$ equals

$\frac{4}{5}$

$\frac{5}{4}$

$\frac{11}{10}$

None of these

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Given $P (A \cup B) = \frac{1}{2}$ and $P (A \cap B) = \frac{2}{5}$
Fact P(AUB) = P(A) + P(B) – P(A⋂B)

$\Rightarrow P(A) + P(B) = P(AUB) + P(A⋂B)$ $= \frac{1}{2} + \frac{2}{5} = \frac{9}{10}$
$1 - P (A^{c}) + 1 - P (B^{c}) = \frac{9}{10}$
$P (A^{c}) + P (B^{c}) = 2 - \frac{9}{10} = \frac{11}{10}$

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