In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.[1] The event A and its complement [not A] are mutually exclusive and exhaustive. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. The complement of an event A is usually denoted as A′, Ac, A or A. Given an event, the event and its complementary event define a Bernoulli 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 two outcomes 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].
^Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Cengage Learning 2007, ISBN 978-0-495-38386-4, p. 229 (restricted online copy, p. 229, at Google Books)
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