# Exhaustive Events

Consider the experiment of throwing a die. We have S = {1, 2, 3, 4, 5, 6}. Let us define the following events

A: 'a number less than 4 appears',

B: 'a number greater than 2 but less than 5 appears' and

C: 'a number greater than 4 appears'.

Then A = {1, 2, 3}, B = {3,4} and C = {5, 6}. We observe that

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

Such events A, B and C are called exhaustive events. In general, if E

_{1}, E

_{2}, ..., E

_{n}are

*n*

events of a sample space S and if

E

_{1}âˆª E

_{2}âˆª E

_{3}âˆª ... âˆª E

_{n}= = S

then E

_{1}, E

_{2}, ...., E

_{n}are called

*exhaustive events*. In other words, events E

_{1}, E

_{2}, ..., E

_{n}are said to be exhaustive if atleast one of them necessarily occurs whenever the experiment is performed.

Further, if E

_{i}

*âˆ©*E

_{j}

*= Ï† for*

*i*â‰

*j*i.e., events E

_{i}

*and E*

_{j}

*are pair wise disjoint and, = S,*

then events E

_{1}, E

_{2}, ..., E

_{n}are called mutually exclusive and exhaustive events.