Question

A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. (b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Answer 1

It is given that a black and a red dice is rolled. When two dices are thrown simultaneously, the total number of outcome is 36, so the number of sample space is 36.

(a)

It is given that a black dice is resulted in a 5.

Let A be the event that sum of the number on dices is greater than 9.

A={ ( 4,6 ),( 5,5 ),( 5,6 ),( 6,4 ),( 6,5 ),( 6,6 ) } P( A )= 6 36 = 1 6

Let B be the event that black dice results in 5.

B={ ( 5,1 ),( 5,2 ),( 5,3 ),( 5,4 ),( 5,5 ),( 5,6 ) } P( B )= 6 36 = 1 6

The common outcomes of the event A and B are,

A∩B={ ( 5,5 ),( 5,6 ) } P( A∩B )= 2 36 = 1 18

The conditional probability of the given event is given by P( A|B ),

P( A|B )= P( A∩B ) P( B ) = 1 18 1 6 = 6 18 = 1 3

Therefore, the conditional probability of obtaining a sum greater than 9, given that black dice is resulted in 5, is 1 3 .

(b)

It is given that the red dice is resulted in a number less than 4.

Let E be the event that sum of the number on dices is 8.

E={ ( 2,6 ),( 3,5 ),( 4,4 ),( 5,3 ),( 6,2 ) } P( E )= 5 36

Let F be the event that red dice results in number less than 4.

F={ ( 1,1 ),( 1,2 ),( 1,3 ),( 2,1 ),( 2,2 ),( 2,3 ), ( 3,1 ),( 3,2 ),( 3,3 ),( 4,1 ),( 4,2 ),( 4,3 ), ( 5,1 ),( 5,2 ),( 5,3 ),( 6,1 ),( 6,2 ),( 6,3 ) } P( F )= 18 36 = 1 2

The common outcomes between the events E and F are,

E∩F={ ( 5,3 ),( 6,2 ) } P( E∩F )= 2 36 = 1 18

The conditional probability of the given event is given by P( E|F ) which is calculated as,

P( E|F )= P( E∩F ) P( F ) = 1 18 1 2 = 2 18 = 1 9

Therefore, the conditional probability of obtaining a sum equal to 8, given that the second dice is resulted in a number less than 4 is 1 9 .