If A and B are two events, the probability that exactly one of them occurs is given by

Engineering

Mathematics

Conditional Probability

Question

$P(A) + P(B) - 2P(A⋂B)$

$P(A⋂B') + P(A' ⋂B)$

$P(A⋂B) - P(A⋂B)$

$P(A') + P(B') - 2P(A' ⋂B')$

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We have P (exactly one of A, B occurs)
$= P(A⋂B') U (A' ⋂B') = P(A⋂B') + P(A' ⋂B)$
$= P(A) - P(A⋂B) + P(B) - P(A⋂B)$
$= P(A) + P(B) - 2P(A⋂B) = P(AUB) - P(A⋂B)$
Also P (exactly one of A, B occurs)
$= [1 - P(A' ⋂B')] - [P(A' UB')]$
$= P(A' UB') - P(A' ⋂B')$
$= P(A') + P(B') - 2P(A' ⋂B')$ .

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