Complementary Events - Examples, Definition, Rule - Cuemath

Complementary Events

Complementary events are two events that exist such that one event will occur if and only if the other does not take place. For two events to be classified as complementary events they must be mutually exclusive and exhaustive.

The sum of probabilities of complementary events must be equal to 1. Complementary events can take place only when there are exactly two outcomes. In this article, we will learn more about complementary events, and see some associated examples

1.	What are Complementary Events in Probability?
2.	Complementary Events Properties
3.	Rule of Complementary Events
4.	Complementary Events Example
5.	FAQs on Complementary Events

What are Complementary Events in Probability?

Complementary events in probability occur when only two outcomes are possible. For example, passing a test or not passing a test. An event can be described as the set of outcomes of an experiment. Thus, events will always be a subset of the sample space.

Complementary Events Definition

Two events can be termed as complementary events if one event can take place only when the other does not occur. Furthermore, the complement of an event can be defined as the set of outcomes in which it does not occur. Let A be an event. The complement of A is denoted as A' or Ac. A and A' together form complementary events.

Complementary Events Properties

For two events to be classified as complementary events they must follow certain properties. These are given as follows:

• Complementary events are mutually exclusive. This means that two events that are complementary cannot occur at the same time. In other words, complementary events are disjoint.
• Complementary events are exhaustive. This implies that an event, as well as its complement, must completely fill up the sample space. Thus, S = A ∪ A'

Rule of Complementary Events

The rule of complementary events states that the sum of probability of occurrence of an event and the probability of occurrence of the complement of that event will always be 1. Let A be an event and P(A) be the probability that A will occur. Thus, P(A') denotes the probability that A will not occur. Then this rule can be mathematically expressed as follows.

P(A) + P(A') = 1

P(A) = 1 - P(A')

P(A') = 1 - P(A)

All these three mathematical statements are equivalent.

Complementary Events Example

Suppose two dice are rolled. Let B be the event of getting two unique digits. Find P(B).

To solve this example, the rule of complementary events will be used as it is an easier approach.

Let B' be the event that the digits on both dice are the same or not unique.

The sample space for B' is S' = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

Total number of outcomes when two dice are rolled = 36

Number of favorable outcomes = 6

P(B') = 6 / 36 = 1 / 6

Using the rule of complementary events, P(A) = 1 - P(A')

P(B) = 1 - (1 / 6) = 5 / 6

Thus, the probability of getting two unique digits on rolling two dice is 5 / 6.

Related Articles:

• Axiomatic Probability Definition
• Experimental Probability
• Probability in Real Life

Important Notes on Complementary Events

• When one event occurs if and only if the other doesn't take place then such events are called complementary events.
• Complementary events are mutually exclusive and exhaustive.
• An event and its complement together form the sample space.
• The rule of complementary events is P(A) + P(A') = 1.