Definition
Mutually Exclusive Events
Two events are said to be mutually exclusive when both the events cannot take place simultaneously. Mutually exclusive events always undergo different outcomes. These events are also known as disjoint events. Mutually exclusive events prevent the second event to take place when the first event appears. For example: If you toss a coin, if it gives you “ Head” then it will not give you “ Tail” at the same time. Hence, they are mutually exclusive.
It is quite essential to differentiate between mutually exclusive events and independent events. Independent events are those which do not depend on each other whereas mutually exclusive events are those that cannot appear at the same time.
For example: If we throw a die then event X {1,4} and Y { 2,5,6} are mutually exclusive
Similarly, if we throw a die then X {1, 4} and Z = {2, 4, 5, 6} are not mutually exclusive because if 4 appears on the die then it will be favorable to both the events X and Z.1
If X and Y happens to be mutually exclusive, then (X n Y) = 0
Complimentary 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.3
Mutually Exclusive Events and Complementary Events
| Aspect | Mutually Exclusive Events | Complementary Events |
| Definition | Events that can never occur together; the occurrence of one event eliminates the possibility of the other event occurring. | Events are complementary if one or the other must occur but they can never occur together. |
| Examples |
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| Why is this important? | If events are mutually exclusive, then you simply add the probabilities of each event. | The probability of complementary events always adds up to 1. You can use the mirror rule to solve some difficult problems. The probability of complementary events occurring simultaneously is 0. |
| How does this affect the general case formula? | P(A and B) = 0 | P(A and B) = 0 P(A or B) = 1 |
Other tips to consider
- Note that the probability of heads and tails in coin tosses is both complementary and mutually exclusive
- All complementary events are, by definition, also mutually exclusive
Mirror Rule Probability
You may be asked the probability of something “not happening” or of “at least one” things happening. By using the complementary rules for probability, we get two equations that make the mirror rule.
P(A happens) = 1 – P(A does not happen)
P(at least one of B happens) = 1 – P(no B happens)
Example
A single coin is tossed 4 times. What is the probability that you will get at least one tail?
Umar, Bobby and Carl S. Pyrdum. Barron’s GMAT. Hauppauge, NY: Barron’s Educational Series, Inc., 2017.
References
1 “Mutually Exclusive Events”. 2022. Vendantu. https://www.vedantu.com/maths/mutually-exclusive-events.
2 “Mutually Exclusive Events”. 2022. Corporate Finance Institute. https://corporatefinanceinstitute.com/resources/knowledge/other/mutually-exclusive-events/.
3 “Complementary Events – Examples, Definition, Rule”. 2022. cuemath. https://www.cuemath.com/data/complementary-events/.
Additional References
Mutually Exclusive Events
“Mutually Exclusive Events – Definition, Formula, Rules, Examples”. 2022. BYJUS. https://byjus.com/maths/mutually-exclusive-events/.
“Mutually Exclusive Events – Explanation & Examples”. 2022. The Story Of Mathematics – A History Of Mathematical Thought From Ancient Times To The Modern Day. https://www.storyofmathematics.com/mutually-exclusive-events.
“Mutually Exclusive Events | Mutually Exclusive Events-Define, Formula, Probability”. 2022. Probability Formula. https://probabilityformula.org/mutually-exclusive-events/.
Complimentary Events
“Complementary Events | Complementary Events – Define, Formula, Solved Examples”. 2022. Probability Formula. https://probabilityformula.org/complementary-events/.
“Complementary Events (Solutions, Examples, Videos)”. 2022. onlinemathlearning.com. https://www.onlinemathlearning.com/complementary-events.html.
What Are Complementary Events In Probability – Solved Problems”. 2022. BYJUS. https://byjus.com/jee/what-are-complementary-events-in-probability/.
Videos
Mutually Exclusive Events
Probability of Mutually Exclusive Events With Venn Diagrams
Mathematical Mysteries 