# 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:

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