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What Are Mutually Exclusive Events? (Mathematics Lesson)

What Are Mutually Exclusive Events?

Two events are mutually exclusive if they cannot occur at the same time.

If I toss a coin, it cannot land on both Heads and Tails at the same time. Getting a Heads and getting a Tails in a coin toss are mutually exclusive events.

Similarly, if I roll a die, I cannot roll a 1 and a 6 at the same time. Rolling a 1 and rolling a 6 are mutually exclusive events.

Probabilities of Mutually Exclusive Events

Mutually exclusive events are useful in probability.

Consider two mutually exclusive events, A and B. It is useful to find the probability of A and B happening, and the probability of A or B happening.
  • 'And'

    Mutually exclusive events cannot both happen.

    The probability of two mutually exlusive events occuring is 0.

    The probability of A and B is 0.

  • 'Or'

    The probability of either mutually exclusive events occuring is found by adding the individual probabilities of each event together.

    The probability of A or B is the probability of A plus the probability of B.

Real Examples of the Probabilities of Mutually Exclusive Events

  • 'And'

    Question: What is the probability of getting a Heads and a Tails in a coin toss?

    Getting a Heads and getting a Tails are mutually exclusive events - getting them both together is impossible. Therefore, the probability of getting both is 0.

    If H is getting a Heads and T is getting a Tails, then:

  • 'Or'

    Question: What is the probability of rolling either a 1 or a 6 on a die?

    Rolling a 1 and rolling a 6 are mutually exclusive events. The probability of getting either a 1 or a 6 is the probility of rolling a 1 plus the probability of rolling a 6.

    Note: The probability of rolling a 1 is ⅙, as is the probability of rolling a 6.

    If 1 is rolling a 1 and 6 is rolling a 6, then:

Visualizing Mutually Exclusive Events on a Venn Diagram

On a Venn Diagram, two mutually exclusive events A and B will be shown as separate, without any intersection.



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Note
WHAT IS PROBABILITY?

Probability tells us how likely something is to happen. Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible.

A probability of 1 means an event is certain.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.



A NOTE ON NOTATION

It is often convenient to use a letter to represent an event.

For example, in a coin toss, let:

  • H be the event getting a Heads, and

  • T be the event getting a Tails
Use P to indicate Probability.

Put brackets after the P, and write the letter for the event inside them.

P(H) means the probability of getting Heads.

P(T) means the probability of getting Tails.

"AND/OR" NOTATION

The notation for the probability of A and B happening is:

P(A ∩ B)

The notation for the probability of A or B happening is:

P(A ∪ B)

MORE THAN TWO MUTUALLY EXCLUSIVE EVENTS

It is possible to have more than two mutually exclusive events. Consider rolling a die. Each possible outcome - getting a 1, 2, 3, 4, 5, or 6 - is mutually exclusive of any other outcome. Therefore there are 6 mutually exclusive events when rolling a die.

To find the probability of an 'or', such as getting a 1 or a 2 or a 3, still involves adding the individual probabilites, only you will have to add three probabilities in this case.

NOT MUTUALLY EXCLUSIVE EVENTS

Many events are not mutually exclusive. It is possible to be male and have brown hair. It is possible for it to be a Monday and for it to be raining outside.