*52*

**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(Aâˆ©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:

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(Aâˆ©B) = 0

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

- P(AâˆªB) = 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(AâˆªB) = P(A) + P(B)
- P(AâˆªB) = 2/6 + 2/6
- P(AâˆªB) = 4/6 = 2/3

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

**Additional Resources**

The following tutorials provide explanations for other common topics in probability:

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