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.

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 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.

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.

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.

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.

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.

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.

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

These are exhaustive as well as equally likely.

X = event of getting a "tail"

Y= event of getting a "head"

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.

M = the events of at least one head

N = the events of at least one tail

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.