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Two terms that students often confuse are **disjoint** and **independent**.

Here’s the difference in a nutshell:

We say that two events are **disjoint** if they cannot occur at the same time.

We say that two events are **independent** if the occurrence of one event has no effect on the probability of the other event occurring.

The following examples illustrate the difference between these two terms in various scenarios.

**Example 1: Flipping a Coin**

**Scenario 1:** Suppose we flip a coin once. If we define event A as the coin landing on heads and we define event B as the coin landing on tails, then event A and event B are **disjoint** because the coin can’t possibly land on heads *and* tails.

**Scenario 2**: Suppose we flip a coin twice. If we define event A as the coin landing on heads on the first flip and we define event B as the coin landing on heads on the second flip, then event A and event B are **independent** because the outcome of one coin flip doesn’t affect the outcome of the other.

**Example 2: Rolling a Dice**

**Scenario 1:** Suppose we roll a dice once. If we let event A be the event that the dice lands on an even number and we let event B be the event that the dice lands on an odd number, then event A and event B are **disjoint** because the dice can’t possibly land on an even number *and* an odd number at the same time.

**Scenario 2**: Suppose we roll a dice twice. If we define event A as the dice landing on a “5” on the first roll and we define event B as the dice landing on a “5” on the second roll, then event A and event B are **independent** because the outcome of one dice roll doesn’t affect the outcome of the other.

**Example 3: Selecting a Card**

**Scenario 1:** Suppose we select a card from a standard 52-card deck. If we let event A be the event that the card is a Spade and we let event B be the event that the card is a Diamond, then event A and event B are **disjoint** because the card can’t possibly be a Spade *and* a Diamond at the same time.

**Scenario 2**: Suppose we select a card from a standard 52-card deck twice in a row with replacement. If we define event A as the card being a Spade on the first draw and we define event B as the card being a Spade on the second draw, then event A and event B are **independent** because the outcome of one draw doesn’t affect the outcome of the other.

**Probability Notation: Disjoint vs. Independent Events**

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

- P(A∩B) = 0

For example, suppose we roll a dice once. Let event A be the event that the dice lands on an even number and let event B be the event that the dice lands on an odd number.

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

- A = {2, 4, 6}
- B = {1, 3, 5}

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.

Thus, we could write:

- P(A∩B) = 0

Similarly, written in probability notation, we say that events A and B are **independent **if the following is true:

- P(A∩B) = P(A) * P(B)

For example, suppose we roll a dice twice. Let event A be the event that the dice lands on a “5” on the first roll and let event B be the event that the dice lands on a “5” on the second roll.

If we write out all of the 36 possible ways for the dice to land, we would find that in only 1 out of the 36 scenarios the dice lands on a “5” both times. Thus, we would say P(A∩B) = 1/36.

We also know that the probability of the dice landing on a “5” during the first roll is P(A) = 1/6.

We also know that the probability of the dice landing on a “5” during the second roll is P(B) = 1/6.

Thus, we could write:

- P(A∩B) = P(A) * P(B)
- 1/36 = 1/6 * 1/6
- 1/36 = 1/36

Since this equation holds true, we could indeed say that event A and event B are **independent** in this scenario.

**Additional Resources**

The following tutorials offer additional information about various statistical terms:

What Are Disjoint Events? (Definition & Examples)

Mutually Inclusive vs. Mutually Exclusive Events