Complementary probability is a popular concept mainly used in statistics. This is derived from the basics of probability theory. The principle of complementary probability is that a set of two complementary events cannot happen at the same time. The concept is studied by not only the students of statistics but also the six sigma professionals as probability is included in their green belt certification courses.
Probability theory is defined as a set of mathematics which deals with random phenomenon. In probability theory, events like E1, E2 En are said to be mutually exclusive wherein the non occurrence of any one of them may lead to n-1 events. Primary objects in probability are events, random variables and stochastic processes. Now the complementary probability deals with various events, which are, to be precise complementary events.
In probability theory, we can mainly classify two events as complementary but one needs to keep in mind that it is called complimentary if and only if one of the possibilities is sure to occur. The events of complimentary probability are mutually exclusive in its own form, e.g. if any one of the two events occurs, the other one, which is the complementary event, cannot occur in any condition. To make the concept, clear we can say that if the probability of an event A, is expressed as P(A), it is not possible for the event to create a probability (also known as complementary event probability) that is to be expressed as 1-P(A). Hence, a set of complementary probability ensures that only one of the events occurs at a time and the other does not take place at the same time.
It is not always mandatory for an event to have a complementary event probability. There are situations where there is no event at all. For an example, if we take a cubical dice, it is not always possible to throw the six from the dice rather six cannot be thrown all the times. But the probabilities of occurrence of the event and also the non occurrence of the event add up to 1. Hence, it can be said that the probability of a six thrown is 1/6 wherein the probability of a six not thrown is 5/6. They add up to 1.
To understand the theory in a better way, we can take the example of the gender of a child which can be either male or female. Thus, we can say that these two events are complementary because Pr (mathym (male) +Pr (mathym (female)) =1. The complement of an event is sometimes known or denoted as A. Another good example to understand complementary probability is calculating the number of smokers and non-smokers in a particular sample of people, picked up randomly. If you know 20% of the samples you have taken are smokers then you can close your eyes and say that the rest of the 80% are non-smokers.
For both the bounded and unbounded random variables, studies are done of the properties. This technique is used to find the series of computation of the sum distribution of uniform random variables which are compounded within and also for the sums of Rayleigh random variables. Presentation of a useful closed form expression of the characteristic of a Rayleigh random variable is also derived. The characteristic function are inverted, a rule mainly trapezoidal for numerical integration and the theorem i.e. the sampling one in the frequency domain are related to and interpreted in terms of this result. The random variables are the properties of probability theory. Generally, there is an event called B so that A and B are both mutually exclusive and exhaustive; that event is the complement of A.
So, the conclusion that can be derived from the above discussion is that the complementary probability is applicable in various events and other factors. But the summation of both the probabilities is always 1 which supports the fact that two complementary events cannot take place simultaneously as this is against the basic principle of the concept.