Complementary Events - Formula & Examples

# Complementary Events

Probability is a chance of prediction of happening of an event. While working with probability we must know about the some terms such as event, sample space, experiment, like events, complementary events and mutually exclusive events etc.

When something happens it is said that an event has occurred. All the events are always associated to some random experiment.

For Example: flipping a coin is an experiment and getting a head or tail are two events which are associated to this experiment.

In probability the complement of any event is the not of that event. In a random experiment, the outcome must occur on every trial and the probabilities of all possible events must total to one.

The probability of complement of an event must be unity minus the probability of the event. For example we have an event B, and then its complement is not B. It is denoted by B$^C$, B', or by a bar over B ($\bar{B}$).

## Formula

The probability of event $B$ is given by $P (B)$ and the probability of its complement by $P (B’)$.

In general this is to remember that the total event of two complementary events is always equal to 1, that is,

$P (B) + P (B’)$ = 1

Thus the probability of complement of event B is given by P (B’) = 1 – P (B).
There are many examples of complementary events associated to different experiments. Some of them are given below:

1) The occurrence of odd or even numbers when we throw a dice.
2) When we flip a coin the occurrence of head and tail are also complementary events.
3) For a given basket of fruits with apples and oranges, picking an apple or orange are again complementary events.
So it can be clearly seen that we have endless number of options for complementary events. To find out the probability of an event also, sometimes we use the complement event probability for easier calculation.

For Example: if we throw a dice 5 times and we need to find the probability of getting a 1 once at least, then using the complement method only can help as if we do it other way we will clearly not get the right answer at all. So we make use of getting no 1 in every trial and then find the complement of it in order to evaluate the required probability.

## Examples

Example 1: A die is rolled. Find the probability of not getting a 5.

Solution: We first find the probability of getting a 5.

P (getting a 5) = $\frac{1}{6}$.

Hence the probability of not getting a five is the complementary event of getting a 5. So the probability is given by:

P (not getting 5) = 1 – P (getting a 5) = 1 – $\frac{1}{6}$ = $\frac{5}{6}$.

Example 2: Out of the six balls given in a bag three are given to be white. Find the probability of not getting a white ball.

Solution: Probability of getting a white ball = P (white) = $\frac{3}{6}$ = $\frac{1}{2}$.

Hence the probability of not getting a white ball = 1 – $\frac{1}{2}$ = $\frac{1}{2}$.

Example 3: We draw a card from a given deck of cards. Find the probability of not getting a jack or king.

Solution: There are 4 jacks and 4 kings so in total we do not require any of these 8 cads to be drawn.

So, probability of getting a jack or king, P (jack or king) = $\frac{8}{52}$ = $\frac{2}{13}$

Hence the probability of not getting a jack or a king is given by:

P (not getting a king or a jack) = 1 – $\frac{2}{13}$ = $\frac{11}{13}$.