probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the complement of any

event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...

''A'' is the event ot ''A'' i.e. the event that ''A'' does not occur.Robert R. Johnson, Patricia J. Kuby: ''Elementary Statistics''. Cengage Learning 2007, , p. 229 () The event ''A'' and its complement ot ''A''are

mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

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′'', ''A^c'',

\neg

''A'' or '. Given an event, the event and its complementary event define a

Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...

: 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

eads Airbus SE (; ; ; ) is a European multinational aerospace corporation. Airbus designs, manufactures and sells civil and military aerospace products worldwide and manufactures aircraft throughout the world. The company has three divisions: '' ...

is logically equivalent to ot tails and ailsis equivalent to ot heads

Complement rule

In a random experiment, the probabilities of all possible events (the

sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...

) must total to 1— that is, some outcome must occur on every trial. For two events to be complements, they must be

collectively exhaustive In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collect ...

, together filling the entire sample space. Therefore, the probability of an event's complement must be

unity Unity may refer to: Buildings * Unity Building, Oregon, Illinois, US; a historic building * Unity Building (Chicago), Illinois, US; a skyscraper * Unity Buildings, Liverpool, UK; two buildings in England * Unity Chapel, Wyoming, Wisconsin, US; a h ...

minus the probability of the event. That is, for an event ''A'', :

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

Equivalently, the probabilities of an event and its complement must always total to 1. This does not, however, mean that ''any'' two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition of

mutual exclusivity In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

Example of the utility of this concept

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 : Pr( 1" on 1st trialor 1" on second trialor ... or 1" on 8th trial := Pr("1" on 1st trial) + Pr("1" on second trial) + ... + P("1" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... 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 the principle of inclusion-exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one "1") = 1 − Pr(no "1"s) := 1 − Pr( o "1" on 1st trialand o "1" on 2nd trialand ... and o "1" on 8th trial := 1 − Pr(no "1" on 1st trial) × Pr(no "1" on 2nd trial) × ... × Pr(no "1" on 8th trial) := 1 −(5/6) × (5/6) × ... × (5/6) := 1 − (5/6)⁸ := 0.7674...

References

External links

''Complementary events''
- (free) page from probability book of

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{{DEFAULTSORT:Complementary Event Experiment (probability theory)

Complement rule

Example of the utility of this concept

See also

References

External links