In probability theory, two events are called complementary if and only if precisely one of the possibilities must occur. Complementary events are mutually exclusive; if one event occurs, then its complement cannot occur. Probability theory is the mathematical study of probability. ... In probability theory, events E1, E2, ... , En are said to be mutually exclusive if the occurrence of any one them automatically implies the non-occurrence of the remaining n-1 events. ...
Definition
Let A and B be two mutually exlusive events. If A and B are complementary events then . If an event A exists, then its complement is sometimes denoted A'. (In the previous example, B is the same as A'.)
Additionally, the probability of complementary events can be found using . This method is useful for finding the probability of an event that is hard to calculate but has a complement that is easier to calculate, e.g. the birthday paradox. The birthday paradox states that if there are 23 people in a room then there is a slightly more than 50:50 chance that at least two of them will have the same birthday. ...
Examples
The gender of a baby involves complementary events. A baby is either born male or female. Thus, these two events are complementary because .
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, and . Thus, and pulling a blue marble and pulling a red marble are complementary events.
An event in probability theory is any fact, which may occur as a result of an experiment with a random outcome or may not.
An event can consist of one or several elementary events, for example, an appearance of two aces one after the other at taking a card out of a pack, or an appearance of the same number at triple throwing of a die.
Such event at throwing of a die is a fall of the die on one of its faces.
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