Question

An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater than 8, B: 2 occurs on either die C: The sum is at least 7 and a multiple of 3. Which pairs of these events are mutually exclusive?

Answer 1

In the given experiment, apair of dice is rolled.

When, a pair of die is rolled, the sample space, S, can be represented as,

S={ ( x,y ):x,y=1,2,3,4,5,6 }

So,

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

Given A: the sum is greater than 8. A={ ( 3,6 ),( 4,5 ),( 4,6 ),( 5,4 ),( 5,5 ),( 5,6 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }

Given B: 2 occurs on either die.

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

Given C: Sum is at least 7 and a multiple of 3,

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

Two events, A and B are mutually exclusive if, A∩B=ϕ.

A∩B=( { ( 3,6 ),( 4,5 ),( 4,6 ),( 5,4 ),( 5,5 ),( 5,6 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }∩ { ( 1,2 ),( 2,1 ), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),( 3,2 ),( 4,2 ),( 5,2 ),( 6,2 ) } ) =ϕ

B∩C={ { ( 1,2 ),( 2,1 ),(2,2),(2,3),(2,4),(2,5),(2,6),( 3,2 ),( 4,2 ),( 5,2 ),( 6,2 ) }∩ { ( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 ),( 6,6 ) } } =ϕ

C∩A=( { ( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 ),( 6,6 ) }∩ { ( 3,6 ),( 4,5 ),( 4,6 ),( 5,4 ),( 5,5 ),( 5,6 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) } ) ={ ( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 ),( 6,6 ) }

Since, A∩B=ϕ and B∩C=ϕ. Hence, the events A and B and events B and C are mutually exclusive.