In probability, exhaustive is a condition of two or more events which serves a great role in finding the probability as it changes if the events are exhaustive or not. Two or more events are said to be exhaustive if there is a certain chance of occurrence of at least one of them when they are all considered together.
Exhaustive events can be either elementary or even compound.  In cases, when we have a single event it depends on the event whether it is exhaustive or not.
For Example: consider the experiment of a fair die being thrown. Then there are six outcomes and all of them are equally likely to occur. Also the events of getting different numbers taken together are exhaustive as together at least one of them is certain to happen. For getting a 2 or 5, sure will get one of the numbers during the experiment. So events are exhaustive.

For if we take events like getting a number less than equal to 3 and a number that is neither composite nor prime… will not be certain to occur and hence are not exhaustive.

Formula

Exhaustive events may or may not be equally likely and mutually exclusive. There is no any particular formula to find the exhaustive events, but sure we are able to tell that events are exhaustive or not after go through below examples.

1) Consider the experiment of throwing a fair die and the event of getting a number less than or equal to 6. Now this particular event is certain in spite of being a single event whenever the die is thrown and hence this single event is also exhaustive.

2) Consider the die throwing experiment. Let the events be getting a number which is multiple of 2 {2, 4, 6} and event of getting factors of 6 {1, 2, 3, 6} and event of getting a {5}. Clearly all these events together are exhaustive.
Suppose we add an event of getting a number that is greater than 3 {4, 5, 6}. Then too, the four events taken together will be exhaustive as we can see in the example here. This is because, when we have something in certain to happen then no matter what we add up to it as new event of that same experiment, but something will still be certain to occur.

Whenever a single event is exhaustive it is always a certain event or we can say that a single event when also certain then it is exhaustive. Whenever we take other events with exhaustive events in combination, we get a new set of exhaustive events only.

Examples

Couple of examples on Exhaustive events are given below:
Example 1: A die is tossed, tell the following events are exhaustive or not.

1) X = Get prime number ; Y = Get multiple of 2 ; Z = Get 1.

2) X = Get prime numbers; Y = Get composite numbers; Z = Get 1.

3) X = an odd number; Y = an even number

Solution:

1) X = Events of getting a prime number = {2, 5, 3}

Y = Events of getting a number that is a multiple of 2 = {4, 2, 6} and

Z = Getting one = {1}

If we toss a coin two of the above mentioned events sure we get. So These are exhaustive together but not mutually exclusive.

2)
X = Events of getting a prime number = {2, 5, 3}

Y = composite number = {4, 6} and

Z = getting one = {1}

Above mentioned events are all together exhaustive and mutually exclusive too.

3) X = events of getting an odd number = {1, 3, 5} and

Y = events of getting an even number = {2, 4, 6}

These are exhaustive as well as equally likely.

Example 2: In the experiment of tossing a coin:

X = event of getting a "tail"

Y= event of getting a "head"

Solution:

X = event of getting a "tail" = {T}

Y= event of getting a "head" = {H}

During experiment at least one of above events will occur.

=> X and Y are exhaustive events.

Example 3: Two coins are tossed. Check whether the events are exhaustive events or not?

M = the events of at least one head

N = the events of at least one tail

Solution:

The event of flipping two coins.

M = the events of at least one head = {HH, HT, TH} and

N = the events of at least one tail = {TH, TT, HT}.

If we are considering only event of at least one head it cannot be exhaustive as it is not certain to happen. But when we take both events together they are certain to happen for any one outcome so the events together are exhaustive.