What Are Disjoint Events? (Definition & Examples)


Disjoint events are events that cannot occur at the same time.

Written in probability notation, events A and B are disjoint if their intersection is zero. This can be written as:

  • P(A and B) = 0
  • P(A∩B) = 0

For example, suppose we select a random card from a deck. Let event A be the event that the card is a Spade or a Club and let event B be the event that the card is a Heart or a Diamond.

We would define the sample space for the events as follows:

  • A = {Spade, Club}
  • B = {Heart, Diamond}

Notice that there is no overlap between the two sample spaces. Thus, events A and B are disjoint events because they both cannot occur at the same time.

Note: Disjoint events are also said to be mutually exclusive.

Examples of Disjoint Events

Here are a few more examples of disjoint events.

Example 1: Coin Toss

Suppose you flip a coin. Let event A be the event that the coin lands on heads and let event B be the event that the coin lands on tails.

Event A and event B would be disjoint because they both cannot occur at the same time. The coin cannot land on heads and tails.

Example 2: Dice Roll

Suppose you roll a dice. Let event A be the event that the dice lands on an odd number and let event B be the event that the dice lands on an even number.

Event A and event B would be disjoint because they both cannot occur at the same time. The dice cannot land on an even number and an odd number.

Example 3: Pro Bowl Location

Suppose the NFL wants to choose a location to host the Pro Bowl. They have narrowed down the options to Miami and San Diego. They place both names in a hat and randomly select one. Let event A be the event that they select Miami and let event B be the event that they select San Diego.

Event A and event B would be disjoint because they both cannot occur at the same time. Miami and San Diego cannot both be selected.

Visualizing Disjoint Events

One useful way to visualize disjoint events is by creating a Venn diagram.

If two events are disjoint then they would not overlap at all in a Venn diagram:

Disjoint events

Conversely, if two events are non-disjoint then there would be at least some overlap in the Venn diagram:

Disjoint vs. Non-disjoint events

The Probability of Disjoint Events

As mentioned earlier, if two events are disjoint then the probability that they both occur at once is zero.

  • P(A∩B) = 0

Similarly, the probability that either event occurs can be calculated by adding up their individual probabilities.

  • P(A∪B) = P(A) + P(B)

For example, let event A be the event that a dice lands on a 1 or a 2 and let event B be the event that a dice lands on a 5 or a 6.

We would define the sample space for the events as follows:

  • A = {1, 2}
  • B = {5, 6}

We would calculate the probability the event A or event B occurs as:

  • P(A∪B) = P(A) + P(B)
  • P(A∪B) = 2/6 + 2/6
  • P(A∪B) = 4/6 = 2/3

The probability that event A or event B occurs is 2/3.

Additional Resources

How to Find the Probability of A or B (With Examples)
How to Find the Probability of A and B (With Examples)
Law of Total Probability: Definition & Examples

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