A pair of dice is rolled once and the sum of the number which appear on the uppermost faces of the two dice is noted. Consider the following events
(i) `E_(1)` : Sum is odd
(ii) `E_(2)` : Sum is 5 or 6
(iii) `E_(3)` : Sum is greater than 11
(iv) `E_(4)` : Sum is less than 4
Find which of the given events are mutually exclusive.

Explanation

Two dice are rolled simultaneously,

It contains `6xx6` = 36 equally likely elementary events.
(i) `E_(1)` = Sum is odd
= {(x, y) `in` S : x + y is odd}
= {(x, y) `in` S : x + y = 3, 5, 7, 9 or 11}

(ii) `E_(2)` : {(x, y) `in` S : x + y = 5 or 6}
= {(1, 4), (2, 3), (3, 2), (4, 1), (1, 5), (2, 4) (3, 3), (4, 2), (5, 1)}
(iii) `E_(3)` : {(x, y) `in` S : x + y `gt` 11}
= {(x, y) `in` S : x + y = 12} = {(6, 6)}
(iv) `E_(4)` : {(x, y) `in` S : x + y `lt` 4}
= {(x, y) `in` S : x + y = 2 or 3}
= {(1, 1), (1, 2) (2, 1)}
Clearly, `E_(1)capE_(2) ne phi, E_(1)capE_(3) = phi, E_(1)capE_(4) ne phi`
`E_(2)capE_(3) = phi, E_(2)capE_(4) = phi " and " E_(3)capE_(4) = phi`
Hence, `E_(1) " and " E_(3), E_(2) " and " E_(3), E_(2) " and " E_(4), E_(3) " and " E_(4)` are mutually exclusive events.