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 A^{c}. 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.
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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.
Examples on Complementary Events

Example 1: Using the rule of complementary events prove that M and N are independent events if P(M_{ }⋃ N) = 1  P(M') P(N').
Solution: P(M_{ }⋃ N) = 1 P(M') P(N')
According to the rule of complementary events, P(A') = 1  P(A)
P(M_{ }⋃ N) = 1  [1  P(M)] [1  P(N)]
P(M_{ }⋃ N) = 1  [1  P(M)  P(N) + P(M).P(N)]
P(M_{ }⋃ N) = 1  1 + P(M) + P(N)  P(M).P(N)
P(M_{ }⋃ N) = P(M) + P(N)  P(M).P(N)
Hence, proved. 
Example 2: There are 10 balls in a bag out of which 3 are black, 2 are red, 1 is blue, 2 are pink, and 2 are purple. Let X be the event of selecting a primary color. Find P(X').
Solution: X = {2 red, 1 blue}
Total balls = 10
Number of favorable outcomes = 3
P(X) = 3 / 10
Using the rule of complementary events, P(A') = 1  P(A)
P(X') = 1  (3 / 10) = 7 / 10
Answer: P(X') = 7 / 10 
Example 3: A random number is chosen from 1 to 55. Calculate the probability of not choosing a perfect square.
Solution: Let Z' be the event of choosing a perfect square. The sample space is given as follows:
Z' = {1, 4, 9, 16, 25, 36, 49}
Total number of outcomes = 50
Favorable outcomes = 7
P(Z') = 7 / 50.
P(Z) = 1  (7 / 50)
= 43 / 50
Answer: P(Z) = 43 / 50.
FAQs on Complementary Events
What is the Meaning of Complementary Events in Probability?
When two events are exhaustive and mutually exclusive they are known as complementary events in probability. Thus, when one event occurs the other cannot take place.
Are Complementary Events Mutually Exclusive?
Yes, complementary events are always mutually exclusive. This means that an event and its complement will not share any outcomes.
Are Complementary Events Disjoint?
Disjoint events are events that cannot occur at the same time. In other words, they are mutually exclusive. Thus, complementary events are disjoint.
What is the Rule of Complementary Events in Probability?
According to the rule of complementary events on adding the probability of an event and the probability of its complement, the result will be 1. In other words, P(A) + P(A') = 1
What is the Difference Between Mutually Exclusive and Complementary Events?
When complementary events are combined they form the sample space. However, this property may not be true for mutually exclusive events. Thus, all complementary events are mutually exclusive but the converse might not be true.
How Do You Tell if Two Events are Complementary Events?
For two events to be complementary events they must follow 3 rules. These are as follows:
 The events are mutually exclusive.
 The events are exhaustive
 P(A) + P(A') = 1
Can Complementary Events be Exhaustive?
All complementary events are exhaustive. This means that the outcomes of the event and its complement together form the sample space.
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