Question

A pair of dice is rolled. Find the probabilities of the given events. (Enter exact numbers as integers, fractions, or decimals.)

(a) The sum is 6.


(b) The sum is 6, given that the sum is less than 8

Answer 1

Solution:

Given: A pair of dice is rolled.

X = Sum of upper most faces when two fair dice are rolled

Outcomes	X = Sum of faces	f : Frequency
(1,1)	2	1
(1,2) , ( 2 , 1)	3	2
(1,3) , (2,2) , ( 3 ,1)	4	3
(1,4) , (2,3) , ( 3 ,2) , (4,1)	5	4
(1,5) , (2,4) , ( 3 ,3) , (4,2) ,(5,1)	6	5
(1,6) , (2,5) , ( 3 ,4) , (4,3) ,(5,2),(6,1)	7	6
(2,6) , (3,5) , ( 4,4) , (5,3) ,(6,2)	8	5
(3,6), (4,5) , (5,4) , (6,3)	9	4
(4,6) , ( 5,5) , ( 6,4)	10	3
(5,6) , (6,5)	11	2
(6,6)	12	1
		N = 36

Part a) Find:

P( The sum is 6)=...........?

P( The sum is 6)= Frequency of 6 / N

P( The sum is 6)= 5 / 36

  

Part b) Find:

P( The sum is 6, given that the sum is less than 8 ) =...........?

P( The sum is 6 | Sum < 8) = ...........?

P( The sum is 6 | Sum < 8) = P( Sum = 6 and Sum < 8) / P( Sum < 8)

P( The sum is 6 | Sum < 8) = P( Sum = 6 ) / P( Sum < 8)

P( The sum is 6 | Sum < 8) = P( Sum = 6 ) / [ P(Sum =2,3,4,5,6,7 ) ]

P( The sum is 6 | Sum < 8) = [ 5/36 ] / [ (1+2+3+4+5+6) / 36 ]

P( The sum is 6 | Sum < 8) = [ 5/36 ] / [ 21/36]

P( The sum is 6 | Sum < 8) = 5/21