**Mutually exhaustive definition:**

A set of event is said to be mutually exhaustive if they include all the possible outcomes of a random experiment taken together. it means if A_{1} , A_{2} ,
A_{3} ,.......... A_{n }are n exhaustive events, then A_{1}`uu`A_{2}`uu` A_{3}`uu` .........`uu` A_{n.}

Example: In the experiment of tossing a coin, let A be the event "Getting Head" and B be the event "Getting Tail"

Then A ={H} and B = {T} then `A uu B = { H , T } = S`

`"Then A and B are mutually exhaustive events"`

`"In probability the mutually exhaustive definition is:"`

a set of events are collectively **exhaustive** if at least one of the events must occur

For example, when rolling a die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes.

So by the mutually exhaustive definition `A uu B =S`

Use the definition of mutually exhaustive events to solve the example

1. In the experiment of rolling a die, let A be the event "getting a **prime number**", B be the event
"getting an even number" and C be the event "getting a number less than 3".

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

`A uu B uu C = { 1 ,2 ,3, 4, 5, 6}`

`"So A, B, and C are exhaustive events."`

`"An experiment involves rolling a pair of dice. "`

`A :"Sum greater than 8"`

`B : "2 appears on either die" `

` "Sum is at least 7 and a multiple of 9"`

`A ={ (3,6) , (4,5) , (4,6) ,(5,4), (5,5) , (5,6) , (6,3), (6,4), (6,5) ,(6,6) }`

`B ={ (2,1) , (2,2) , (2,3) ,(2,4), (2,5) , (2,6) , (1,2), (3,2), (4,2) ,(5,2) ,(6,2) }`

`C ={ (3,6) , (4,5) , (5,4) , (6,3),(6,6) }`

` A uu B != S`

`"So A and B are not mutually exhaustive"`

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**Some properties based on the definition of mutually exhaustive events:**

1.If A and b are two mutually exclusive and mutually exhaustive events then

` P(A) + P(B) = 1`

2. Exhaustive events may be elementary or compound events. They may be **equally likely or not equally likely**`.`

3. A **single event** forms an exhaustive event if it happens for sure whenever the experiment is conducted.

4. Where one or more events are already exhaustive, any other events combined together would always be exhaustive.

5. The event **exhaustive events** and **mutually exclusive** events are independent of each other.

6. The **exhaustive events** and **not equally likely **events are independent of each other.

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The events are said to be not exhaustive when they are such that exist at least one elementary event in the experiment that does not form a part of those events taken together.

For example:In the experiment of tossing a coin, consider A : the event of getting a head, and B : the event of getting a tail. If only A is taken it is Not/non Exhaustive.

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