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A set of events is **collectively exhaustive** if at least one of the events *must* occur.

For example, if we roll a die then it must land on one of the following values:

- 1
- 2
- 3
- 4
- 5
- 6

Thus, we would say that the set of eventsÂ **{1, 2, 3, 4, 5, 6}** is **collectively exhaustive** because the die *must* land on one of those values.

In other words, that set of events, as a *collection*, *exhausts* all possible outcomes.

The following examples show some more situations that illustrate collectively exhaustive events:

**Example 1: Flipping a Coin**

Suppose we flip a coin one time. We know that the coin must land on one of the following values:

- Heads
- Tails

Thus, the set of eventsÂ **{Heads, Tails}** would be collectively exhaustive.

**Example 2: Spinning a Spinner**

Suppose we have a spinner that has three different colors: red, blue and green.

If we spin it one time then it must land on one of the following values:

- Red
- Blue
- Green

Thus, the set of eventsÂ **{Red, Blue, Green}** would be collectively exhaustive.

However, the set of eventsÂ **{Red, Green}** would *not* be collectively exhaustive because it does not contain all possible outcomes.

**Example 3: Types of Basketball Players**

Suppose we have a survey that asks individuals to select their favorite basketball player position. The only potential responses are:

- Point Guard
- Shooting Guard
- Small Forward
- Power Forward
- Center

Thus, the set of eventsÂ **{Point Guard, Shooting Guard, Small Forward, Power Forward, Center}** would be collectively exhaustive.

However, the set of eventsÂ **{Point Guard, Shooting Guard, Small Forward}** would *not* be collectively exhaustive because it does not contain all possible outcomes.

**The Importance of Collectively Exhaustive Events in Surveys**

When designing surveys, itâ€™s particularly important that the responses to the questions are collectively exhaustive.

For example, suppose a survey asks the following question:

**What is you favorite basketball player position?**

And suppose the potential responses were:

- Point Guard
- Shooting Guard
- Small Forward
- Power Forward

Since the positionÂ *Center* was left out, these responses are not collectively exhaustive.

This means that someone who prefers *Center* as their favorite position will have to pick one of the other options, which means the responses to the survey wonâ€™t reflect the true opinions of the respondents.

**Collectively Exhaustive vs. Mutually Exclusive**

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

For example, let event A be the event that a die lands on an even number and let event B be the event that a die 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, which means theyâ€™re mutually exclusive. They also happen to be collectively exhaustive because combined theyâ€™re able to account for all the potential outcomes of the die roll.

However, suppose we define event A and event B as follows:

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

In this case, there is some overlap between A and B so they are not mutually exclusive. However, combined theyâ€™re still able to account for all the potential outcomes of the die roll.

This illustrates an important point:Â **A set of events can be collectively exhaustive without being mutually exclusive**.