1213 LearnersLast updated on 9 September 2025
Exhaustive Events
Exhaustive events are specific events within a sample space such that at least one of them is certain to occur. In other words, in a sample space, there are some sets of events where only one set of the event can occur out of all.
What are Exhaustive Events?
Exhaustive events are events where one event among all the possible outcomes is certain to happen. Here, events refer to all possible outcomes in a probability experiment.
Let’s suppose a sample space S of rolling a die in an experiment and let event A of getting an even number be E1= {2, 4, 6}.
Event B of getting an odd number E2 = {1, 3, 5}.
Since every number from 1 to 6 are either odd or even, event A and B together cover all outcomes, union of the two exhaustive events will give out the sample space.
A and B are exhaustive events, but only one of these events can occur.
There are two types of exhaustive events, they are mutually exclusive exhaustive events and exhaustive events that are not mutually exclusive.
Difference between Mutually Exclusive and Exhaustive Events
Mutually exhaustive events or mutually exclusive collectively exhaustive events are exhaustive events that are mutually exclusive. Mutually exclusive here means, events that cannot happen at the same time. Let’s understand them in detail using a table.
| Feature | Mutually Exclusive Events | Exhaustive Events |
| Definition | Events that cannot happen at the same time. |
It is a set of events that together cover all possible outcomes.
|
| Mathematical Condition | A ∩ B = ∅ (No common outcomes). | A ∪ B ∪ C = S(Covers the sample space). |
| Overlap |
No overlap (if one happens, the other cannot). |
May or may not overlap. |
| Example (Coin Flip) |
"Getting Heads" and "Getting Tails" are mutually exclusive because both cannot occur together.
|
"Getting Heads" and "Getting Tails" are also exhaustive because they cover all possible results. |
| Example (Dice Roll) | "Getting an even number" ({2,4,6}) and "Getting an odd number" ({1,3,5}) are mutually exclusive. |
"Getting a number ≤ 3" ({1,2,3}) and "Getting an even number" ({2,4,6}) are exhaustive because their union covers {1,2,3,4,5,6}.
|
| Example (Cards) | "Drawing a heart" and "Drawing a spade" are mutually exclusive. |
"Drawing a face card" and "Drawing a number card" are exhaustive because they cover all cards in the deck.
|
How to Represent Exhaustive Events in a Venn Diagram?
In order to represent the exhaustive events more clearly, graphical representation using Venn diagrams can be of a great help to the students learning about probability, events, its types etc. Here is the Venn representation of the exhaustive events.
Let’s take an example, in a sample space of rolling a die S = {1, 2, 3, 4, 5, 6}. Event A of getting an even number on the die EA = {2, 4, 6}. Event B of getting an odd number on the die EB = {1, 3, 5}. Event C of getting a prime number on the die EC = {2, 3, 5}.
Here, some numbers of the dice exist in both the events. So this event can be termed as an exhaustive event, but not mutually exclusive. Because mutual exclusive events should not overlap the numbers with other events of the same set.
1. First, find all the possible outcomes of a given event.
2. Find the outcomes which overlap with other events (if possible, otherwise it will be a mutually exclusive event).
3. Draw circles with the event that includes both outcomes of the other events placed at the center. Since Event C has both the outcomes of Event A and Event B, Event C will overlap both the other events and represent in the center.
4. Event C = {2, 3, 5}; numbers 2 is in Event A, and 3, 5 overlap with Event B. Now write those outcomes in the overlapping part of the two circles.
5. Repeat the process for all the events.
Real Life Applications of Exhaustive Events
Exhaustive events can be found in real life situations. Let’s analyze some of the real-life applications of exhaustive events.
- Traffic Light System: There are events like red light, green light, or yellow light. These are all in the sample space of traffic lights. But only one of these events can occur.
- Student's Exam Result: During examination results, a student can be marked as either pass or fail, and not both of them at the same time.
- Election Results: Election results decide which candidate wins. Making it exhaustive, as the event possibilities are “candidate A wins”, or “candidate B wins” or “Tie” (re-election). The events here cannot happen together.
Common Mistakes of Exhaustive Events and How to Avoid Them
Making mistakes are common in exhaustive events, especially when learners are new to this topic. Here are some of a few common mistakes that students might make and how to avoid them.
Mistake 1
Thinking that exhaustive events must be mutually exclusive.
Exhaustive events only mean that they cover all possible outcomes. They can overlap. Check if their union equals the sample space.
Mistake 2
Forgetting to include all possible outcomes.
Always list out the full sample space first, then check if the given events cover every outcome.
Mistake 3
Believing that the overlapping events are always exhaustive.
Just because the events overlap doesn’t mean they cover all the outcomes. To be exhaustive, their union must equal the entire sample space.
Mistake 4
Assuming a single event can be exhaustive.
No single event can be exhaustive unless it includes all possible outcomes. Exhaustive events are usually a group of events.
Mistake 5
Confusing exhaustive events with equally likely events.
Exhaustive events cover all possible outcomes, but they do not have to equally likely. Check probabilities separately to avoid confusion.
Solved Examples of Exhaustive Events
Problem 1
A bag contains red, blue, and green balls. If an event is defined as picking a red ball and another event as picking a blue or green ball, are these events exhaustive?
Yes, because together they cover all possible outcomes.
Explanation
The sample space includes (Red, Blue, Green).
The events are Event A: Picking a red ball = Red
Picking a blue or green ball = Blue, green
Since A ⋃ B = Red, Blue, Green = Sample Space, these are exhaustive events.
Problem 2
When flipping a coin, are the events “getting heads” and “getting tails” exhaustive?
Yes, because they include all possible outcomes of the coin flip.
Explanation
The sample space is (Heads, Tails). The two events are:
Event A: Getting heads = {H}
Event B: Getting tails = {T}
Since A ⋃ B = {HT}, they cover all possibilities, making them exhaustive events.
Problem 3
When rolling a six-sided die, are the events “getting an even number” and “getting an odd number” mutually exclusive?
Yes, because no number can be both even and odd at the same time.
Explanation
Event A: Getting an even number = {2, 4, 6}
Event B: Getting an odd number = {1, 3, 5}
Since A ⋂ B = Ø (no common outcomes), these events are mutually exclusive.
Problem 4
A die is rolled. Are the events “getting a prime number” and “getting an odd number” mutually exclusive?
No, because some numbers are both prime and odd.
Explanation
Event A (Prime numbers): {2, 3, 5}
Event B (Odd numbers): {1, 3, 5}
The common numbers are {3, 5}, so A ∩ B
FAQs Related to Exhaustive Events
1.Can exhaustive events be mutually exclusive?
2.What is the difference between exhaustive and mutually exclusive events?
3.Are all sets of events exhaustive?
4.Can an event be exhaustive by itself?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
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