Exhaustive Events

Exhaustive events are a series of occurrences in a sample space that occur compulsorily while doing the experiment. In layman’s terms, all potential occurrences in a sample space of an experiment represent exhaustive events. For example, there are two conceivable outcomes when flipping an impartial coin: heads or tails. As a result, these two possibilities are exhaustive occurrences since one of them will undoubtedly occur when flipping the coin.

In this post, we will look at the notion of exhaustive occurrences, its definition, and its likelihood. We’ll also go through mutually exclusive exhaustive events with examples to help you grasp the distinction between mutually exclusive exhaustive and exhaustive events.

What Is Meant By Exhaustive Events

Because one of the potential outcomes of an experiment will undoubtedly occur, all conceivable outcomes comprise exhaustive occurrences. Exhaustive occurrences may or may not be equally likely events, i.e., events do not have to be equally likely to be exhaustive. Consider an example of an exhaustive occurrence. When rolling a dice, there are six possible outcomes: 1, 2, 3, 4, 5, 6. If we roll a dice, one of these six possibilities will almost certainly occur. As a result, all six outcomes are exhaustive occurrences. As a result, the union of the exhaustive events yields the complete sample space.

Let us substitute rolling dice for the occurrences and see if the events are exhaustive and compose the sample space. Let A represent the event of obtaining a prime number, B represent the event of getting a composite number, and C represents the event of getting the number 1. We now have A ={ 2, 3, 5} B ={ 4, 6} and C ={1}. When we roll a die, one of the six numbers – 1, 2, 3, 4, 5, 6 – will appear, implying that one of the occurrences A, B, or C will occur. As a result, they are comprehensive events. A U B U C also equals {1, 2, 3, 4, 5, 6} = Sample Space.

If there are n events in a sample space
S 

E1 , E2,E3,….En

and if

E1  E2   E3  …..  En = Uni=1 Ei= S

then

E1 , E2,E3,….En

Are known as exhaustive events.

In other words, exhaustive occurrences occur if at least one of them must occur whenever the experiment is carried out.

In Probability, what are Exhaustive Events?

Consider the experiment of rolling a dice.

S = 1, 2, 3, 4, 5, 6 sample space

Assume that the events linked with this experiment are A, B, and C. Let us also characterise these occurrences as follows:

A is the occurrence of receiving a number larger than three.

B is the occurrence of receiving a number larger than 2 but less than 5.

C denotes the occurrence of receiving a number fewer than three.

These occurrences can be written down as follows:

A = 4, 5, and 6

3 + 4 = B

as well as C = 1, 

We notice that

A ⋃ B ⋃ C ={ 4, 5, 6} = {3, 4} ={ 1, 2} ={ 1, 2, 3, 4, 5, 6} = S

As a result, A, B, and C are referred to as exhaustive events.

The probability of exhaustive events, on the other hand, is one. 

Events That Are Mutually Exhaustive

Mutually exclusive exhaustive occurrences are referred to as mutually exhaustive events or mutually exclusive collectively exhaustive events (MECE). Such occurrences cannot occur more than once at the same moment, and at least one will occur anytime the experiment is carried out. Some frequent instances of mutually exclusive occurrences include:

• When a die is rolled, the set of all possible six outcomes 1, 2, 3, 4, 5, 6 is mutually exhaustive since no two numbers can occur at the same time and one of them will always appear.
• When flipping a coin, there are two possible outcomes: heads or tails. These two occurrences are mutually exclusive since they cannot occur concurrently and they both form the  Sample Space.

Let us now look at an example of a non-exclusive exhaustive event. Consider the following experiment: rolling a dice. Let A represent the event of receiving a prime number, B represent the event of receiving an even number, and C represents the event of receiving an odd number. This means that A ={ 2, 3, 5,} B ={ 2, 4, 6,} and C ={ 1, 3, 5}. These three occurrences encompass all conceivable outcomes in this case. A, B, C, i.e., A U B U C = Sample space, but these events have common outcomes., i.e., A U B U C = Sample space, but these events have common outcomes.

A ∩ B = {2} A ∩ C = {3,5}

A, B, and C are not mutually exclusive occurrences, but they are exhaustive.

CONCLUSION:-

Mutually exhaustive events occur when a sample space S is partitioned into multiple mutually exclusive events, the union of which comprises the whole sample space. The likelihood of an exhaustive event occurring is always one. The intersection of mutually exclusive exhaustive events is always null and void.