Exhaustive Events

Exhaustive events are a set of events in a sample space such that one of them compulsorily occurs while performing the experiment. In simple words, we can say that all the possible events in a sample space of an experiment constitute exhaustive events. For example, while tossing an unbiased coin, there are two possible outcomes - heads or tails. So, these two outcomes are exhaustive events as one of them will definitely occur while flipping the coin.

Let us understand the concept of exhaustive events in this article, its definition, and probability. We will also discuss the mutually exclusive exhaustive events with examples to understand the difference between mutually exclusive exhaustive and exhaustive events.

All possible outcomes of an experiment constitute exhaustive events as one of them will definitely occur. Now, exhaustive events may or may not be equally likely events, i.e., it is not necessary for events to have equal probability to be exhaustive. Let us consider an example of exhaustive events. There are six possible outcomes when rolling a die which is {1, 2, 3, 4, 5, 6}. Now, if we roll a die, one of these six outcomes will definitely occur. Hence, all these six outcomes are exhaustive events. Therefore, we can say that the union of the exhaustive events gives the entire sample space.

Let us change the events for rolling a die and verify if the events are exhaustive and constitute the sample space. When rolling a die, let A be the event of getting a prime number, let B be the event of getting a composite number and C be the event of getting the number 1 (as 1 is neither prime nor composite and it is one of the possible outcomes). Now, we have A = {2, 3, 5}, B = {4, 6}, C = {1}. Now, when we roll a die, one of the six numbers - 1, 2, 3, 4, 5, 6 will occur which implies one of the events A, B, C will occur. Hence, these are exhaustive events. Also, A U B U C = {1, 2, 3, 4, 5, 6} = Sample Space.

Consider a sample space S for an experiment and let A_i be a possible event for all i ∈ {1, 2, 3, 4, ..., n} in S while performing the experiment. Then, A_i is said to be an exhaustive event if at least one of the A_is occurs whenever the experiment is performed and \(\bigcup_{i=1}^{n}A_i = S\).

It is not necessary for exhaustive events to be mutually exclusive. In other words, we can say that exhaustive events are of two types - mutually exclusive exhaustive events and exhaustive events that are not mutually exclusive. Let us now see the Venn diagram for exhaustive events in a sample space to understand the concept.

Now, given below is a diagram showing the entire sample space of an experiment and A_i for i ∈ {1, 2, 3, 4, 5} are the possible events in S. Now, A₁, A₂, A₃, A₄, A₅, are exhaustive events as one of them will definitely occur whenever the experiment is performed. Please note that the diagram shows exhaustive events that are not mutually exclusive as these events intersect with each other which implies more than one experiment can occur simultaneously.

Exhaustive events that are mutually exclusive are said to be mutually exhaustive events or mutually exclusive collectively exhaustive events (MECE). Such events cannot occur more than one at the same time and one of them will definitely occur whenever the experiment is performed. Some of the common examples of mutually exhaustive events are:

• When rolling a die, the set of all possible six outcomes {1, 2, 3, 4, 5, 6} is mutually exhaustive as no two numbers can appear at the same time and one of them will definitely appear.
• When tossing a coin, the two possible outcomes are heads and tails. These two events are mutually exhaustive events as they cannot occur at the same time and they both constitute the sample space.

Now, let us see an example of an exhaustive event that is not mutually exclusive. Consider an experiment of rolling a die. Let A be the event of getting a prime number, B be the event of getting an even number and C be the event of getting an odd number. This implies we have A = {2, 3, 5}, B = {2, 4, 6}, C = {1, 3, 5}. Here all possible outcomes are covered by these three events A, B, C, i.e., A U B U C = Sample space but these events have common outcomes., i.e.,

A ∩ B = {2}, A ∩ C = {3, 5}

⇒ A, B, C are not mutually exclusive events but these are exhaustive events.

Given below, we have a diagram showing the entire sample space of an experiment and A_i for i ∈ {1, 2, 3, 4, 5} are the possible events in S. Now, A₁, A₂, A₃, A₄, A₅, are mutually exhaustive events as one of them will definitely occur whenever the experiment is performed and no two of them can occur simultaneously.

Important Notes on Exhaustive Events

• When a sample space S is divided into many mutually exclusive events such that their union forms the entire sample space, these events are said to be mutually exhaustive events.
• The probability that an exhaustive event will occur is always 1.
• The intersection of mutually exclusive exhaustive events is always empty.

Articles Related to Exhaustive Events

• Bernoulli Trials
• Multiplication Rule of Probability
• Probability Rules

What are Exhaustive Events in Probability?

The set of events out of which one will definitely occur whenever the experiment is performed is called exhaustive events in probability. The union of the exhaustive events gives the entire sample space.

What are Mutually Exhaustive Events?

When a sample space S is divided into many mutually exclusive events such that their union forms the entire sample space, these events are said to be mutually exhaustive events. Hence, exhaustive events that are mutually exclusive are said to be mutually exhaustive events or mutually exclusive collectively exhaustive events (MECE).

What are Collectively Exhaustive Events?

The collection of exhaustive events such that one of the events from the collection must occur whenever the experiment is performed is called the jointly or collectively exhaustive events.

What is the Difference Between Mutually Exclusive and Exhaustive Events?

Mutually exclusive events are the events in probability such that two events can occur at the same time. On the other hand, exhaustive events are a set of events in a sample space such that one of them compulsorily occurs while performing the experiment. Exhaustive events may or may not be mutually exclusive.

What is the Difference Between Exhaustive Events and Sample Space?

The union of exhaustive events of an experiment forms the sample space. This implies exhaustive events are subsets of the sample space as the sample space contains the exhaustive events.

Are all Exhaustive Events Mutually Exclusive?

No, all exhaustive events are not mutually exclusive. For example, let A be the event of getting a prime number, B be the event of getting an even number and C be the event of getting an odd number. This implies we have A = {2, 3, 5}, B = {2, 4, 6}, C = {1, 3, 5}. Here all possible outcomes are covered by these three events A, B, C, i.e., A U B U C = S but these events have common outcomes., i.e., A ∩ B = {2}, A ∩ C = {3, 5} ⇒ A, B, C are not mutually exclusive events but these are exhaustive events.