1255 LearnersLast updated on June 4th, 2025
Complementary Events
Complementary events are the opposite outcomes of a given event, which means an event will occur if and only if the original event does not occur. They are mutually exclusive and together cover all possible outcomes. In this topic, we are going to learn about complementary events and their properties.
What are complementary events?
A pair of events where one happens if and only if the other event does not happen is called a pair of complementary events. Complementary events are mutually exclusive (cannot happen at the same time) and exhaustive (cover all outcomes).
If we take an event A, its complement A’ or Ac consists of all outcomes that are not in A. An example would be flipping a coin. Here, heads would be Event A and tails would be Event Ac.
To know whether two events are complementary or not, you need to know a few of these properties:
- Complementary events are mutually exclusive, which means they cannot occur simultaneously. If one event occurs, then the other event cannot occur at the same time.
- Complementary events cover all outcomes in the sample space. So either event A or its complementary event must happen.
What is the Rule of Complementary Events?
In complementary events, the sum of probabilities is always equal to 1. If the probability of event A is P(A), the probability of event Ac is P(Ac) = 1 - P(A). We call this the probability sum rule.
Mathematically, it is expressed as:
P(A’) = 1 - P(A)
P(A) = 1 - P(A’)
P(A) + P(A’) = 1
These three statements are all equivalent.
Where:
P(A’): is the probability of complementary event A’
P(A): is the Probability of event A
Real-life applications on Complementary events
Complementary events are events that occur only if the other event does not occur. Here are a few real-world applications of complementary events.
- Medical testing: When testing a patient for a disease, they would either have the disease or not. They will not be able to have both.
- Sports outcomes: In a match, a team will either win, lose or tie.
- Airline safety: A flight will either arrive on time or will be delayed.
Common Mistakes and How to Avoid Them in Complementary Events
Students tend to make mistakes when learning about complementary events. Here are some mistakes that students make and ways to avoid them:
Mistake 1
Getting confused between complementary and mutually exclusive events
Students assume that all mutually exclusive events are complementary; this is wrong. Complementary events must cover all possible outcomes, which mutually exclusive events do not need to.
Solved examples on Complementary events
Problem 1
If we flip a coin, what is the probability of not getting heads?
P(not heads) = 0.5
Explanation
Since the probability of getting heads is 0.5, the complement (not getting heads) is: 1 - 0.5 = 0.5.
Problem 2
A six-sided die is rolled. What is the probability of not rolling a 3
P(not rolling a 3) = 5/6
Explanation
The probability of rolling a 3 is 1/6, so the probability of not rolling a 3 is:
1 - 1/6 = 5/6.
Problem 3
If a student has a 0.85 probability of passing an exam, what is the probability of the student failing?
P(failing) = 0.15
Explanation
The probability of the student passing is 0.85
So, the probability of a student failing is 1 - 0.85 = 0.15
Problem 4
An online store guarantees that 92% of orders arrive on time. What is the probability of an order being delayed?
P(delayed) = 0.08
Explanation
Since P(on time) = 0.92 (92/100)
Complement = 1 - 0.92 = 0.08
Problem 5
A machine in a factory has a -.15 probability of breaking down in a month. What is the probability that it does not break down?
P(no breakdown) = 0.85
Explanation
The probability of breaking down is 0.15
Complement = 1 - 0.15 = 0.85.
FAQs on Complementary events
1.How to calculate the probability of complement events?
2. Can complementary events happen at the same time?
3.What is the difference between complementary events and mutually exclusive events?
4.What will happen if an event does not cover all outcomes?
5.Why must the sum of the probability of complementary events be equal to 1?
6.How can children in Saudi Arabia use numbers in everyday life to understand Complementary Events?
7.What are some fun ways kids in Saudi Arabia can practice Complementary Events with numbers?
8.What role do numbers and Complementary Events play in helping children in Saudi Arabia develop problem-solving skills?
9.How can families in Saudi Arabia create number-rich environments to improve Complementary Events skills?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
