Complementary event
Encyclopedia
In probability theory
, the complement of any event
A is the event [not A], i.e. the event that A does not occur. 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
, 
or 
.
It may be tempting to say that
That 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.
Instead one may find the probability of the complementary event and subtract it from 1, thus:
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the complement of any event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...
A is the event [not A], i.e. the event that A does not occur. The event A and its complement [not A] are mutually exclusive
Mutually exclusive
In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both....
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
, or .Simple examples
- A coin is flipped and one assumes it cannot land on its edge. It can either land on "heads" or on "tails" Because these two events are complementary, we have
- Three plastic balls are in a bag. One is blue and two are red. Assuming that each has an equal chance of being pulled out of the bag,
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 trial] or ["1" on second trial] or ... 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... (...and this is clearly wrong.)
That 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.
Instead one may find the probability of the complementary event and subtract it from 1, thus:
- Pr(at least one "1") = 1 − Pr(no "1"s)
- = 1 − Pr([no "1" on 1st trial] and [no "1" on 2nd trial] and ... and [no "1" on 8th trial])
- = 1 − Pr(no "1" on 1st trail) × 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...