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Union, intersection and complement

For two events $A$ and $B$ , the union rule gives the probability that at least one of them occurs; the intersection $A\cap B$ must be subtracted so it is not counted twice. The complement $A^c$ is useful for "at least one" questions, since it is often easier to compute the opposite event.

P(A\cup B) = P(A) + P(B) - P(A\cap B)

probability of

A

or

B

P(A^c) = 1 - P(A)

the complement rule

Symbols

$A\cup B$	$A$ or $B$ (or both)
$A\cap B$	$A$ and $B$ together
$A^c$	the complement of $A$ (not $A$ )

Example

$P(A) = 0.45$ , $P(B) = 0.65$ , $P(A\cap B) = 0.15$ :

$P(A\cup B) = 0.45 + 0.65 - 0.15 = 0.95$ .

"At least one of

n

" is often easiest to find as

1 -

the probability that none occur.

Practise probability for free →

Series and parallel systems →

Part of Statistics and Risk Analysis: Probability.