Mutually Exclusive Events: Definition and Examples

If two events are mutually exclusive, it means that they cannot occur at the same time. For example, the two possible outcomes of a coin flip are mutually exclusive; when you flip a coin, it cannot land both heads and tails simultaneously.

Coin flips are mutually exclusive.

The outcomes of a coin flip are mutually exclusive; a coin cannot land both heads and tails simultaneously.
Photo by Public Domain Pictures.

By contrast, rain and sunshine are not mutually exclusive; while rare, it is possible to have a sunshower, when it rains while the sun still shines.

rain and sunshine are non-mutually exclusive.

Rain and sunshine are not mutually exclusive (that is, they can occur together), as shown by this image of a sunshower. Photo by Wikimedia Commons.

Examples with playing cards

Let’s look at a few examples of mutual exclusivity involving playing cards. If we have a single standard deck of 52 cards, which of the following pairs of events are mutually exclusive?

A) Drawing a red card or drawing a king
B) Drawing a red card or drawing a club
C) Drawing a black card or drawing a spade
D) Drawing a black card or drawing an ace

mutual exclusivity with playing cards

Standard playing cards. Photo by Pixabay.

The correct answer is B). Drawing a red card and drawing a club cannot occur at the same time, because all clubs are black; therefore, the outcomes drawing a red card and drawing a club are mutually exclusive.

The other scenarios are not mutually exclusive, because the two characteristics listed can occur together.

  • A) a card can be both red and a king (the king of hearts or the king of diamonds)
  • C) a card can be both black and a spade (all spades in the deck are black)
  • D) a card can be both black and an ace (the ace of spades or the ace of clubs).
  •  
    For a more formal treatment of mutual exclusivity involving set theory and more involved practice problems, check out this page from the University of California at Berkeley. Otherwise, continue on for a few more examples!

    Mutually exclusive events with a standard 6-sided die

    Now that we have a framework for mutually exclusive events, let’s try a few more examples, this time with a standard 6-sided die. Let’s imagine we are rolling this die just once. Can you identify the following pairs of events as mutually exclusive or non-mutually exclusive?

    1) Rolling a number divisible by 2 or rolling a number divisible by 3
    2) Rolling a number divisible by 2 or rolling a number that is a multiple of 5
    3) Rolling a prime number or rolling an even number
    4) Rolling a non-prime number or rolling an odd number

    Answers:
    1) Non-mutually exclusive (you could a roll a 6, which is divisible by both 2 and 3)
    2) Mutually exclusive (you cannot roll a 2,4, or 6 at the same time as you roll a 5)
    3) Non-mutually exclusive (you could roll a 2, which is an even prime number)
    4) Mutually exclusive (the only non-prime numbers on the die are 4 and 6, which are not odd)

    Need more practice identifying mutually exclusive events? Check out our statistics lessons and videos!

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