| Part of a series onstatistics |
| Probability theory |
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Inprobability theory, thecomplement of anyeventA is the event [not A], i.e. the event thatA does not occur.[1] The eventA and its complement [not A] aremutually exclusive andexhaustive. Generally, there is only one eventB such thatA andB are both mutually exclusive and exhaustive; that event is the complement ofA. The complement of an eventA is usually denoted asA′,Ac,A orA. Given an event, the event and its complementary event define aBernoulli trial: did the event occur or not?
For example, if a typical coin is tossed and one assumes that it cannot land on its edge, then it can either land showing "heads" or "tails." Because these twooutcomes are mutually exclusive (i.e. the coin cannot simultaneously show both heads and tails) and collectively exhaustive (i.e. there are no other possible outcomes not represented between these two), they are therefore each other's complements. This means that [heads] is logically equivalent to [not tails], and [tails] is equivalent to [not heads].
In arandom experiment, the probabilities of all possible events (thesample space) must total to 1— that is, some outcome must occur on every trial. For two events to be complements, they must becollectively exhaustive, together filling the entire sample space. Therefore, the probability of an event's complement must beunity minus the probability of the event.[2] That is, for an eventA,
Equivalently, the probabilities of an event and its complement must always total to 1. This does not, however, mean thatany two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition ofmutual exclusivity.
Suppose one throws an ordinary six-sided die eight times. What is the probability that one sees a "1" at least once?
It may be tempting to say that
This result cannot be right because a probability cannot be more than 1. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive.
One may resolve this overlap by theprinciple of inclusion-exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: