Question

Find the probability that the sum is as stated when a pair of dice is rolled. (Enter your answers as fractions.) (a) odd and less than 5 (b) odd or less than 5

Answer 1

Solution:

We are given that: two dice are rolled.

Thus sample space of rolling of two dice is:

X = sum of numbers obtained on uppermost faces of two dice

then following is the table which gives all possible values of sum of numbers and their frequencies.

For example: (1,1) gives sum = 1+1=2 ,

(1,2) gives sum = 1+2=3

and so on.

We get:

Die 1	Die 2	Sum
1	1	2
1	2	3
1	3	4
1	4	5
1	5	6
1	6	7
2	1	3
2	2	4
2	3	5
2	4	6
2	5	7
2	6	8
3	1	4
3	2	5
3	3	6
3	4	7
3	5	8
3	6	9
4	1	5
4	2	6
4	3	7
4	4	8
4	5	9
4	6	10
5	1	6
5	2	7
5	3	8
5	4	9
5	5	10
5	6	11
6	1	7
6	2	8
6	3	9
6	4	10
6	5	11
6	6	12

Thus frequency distribution is:

X = Sum of Numbers	Frequency
2	1
3	2
4	3
5	4
6	5
7	6
8	5
9	4
10	3
11	2
12	1
	N = 36

Part a) P( Sum is odd and less than 5 ) =................?

From above frequency table, we can see: Sum less than 5 are ( 2,3,4) and out of which odd sum is only 3

and sum =3 has frequency 2

Thus we have to find:

P( Sum is odd and less than 5 ) = P( Sum = 3)

P( Sum is odd and less than 5 ) = 2 / 36

P( Sum is odd and less than 5 ) = 1/18

Part b) P(Sum is odd or less than 5 )=........?

P(Sum is odd or less than 5 )=P( Sum is Odd ) + P( Sum is less than 5) - P( Sum if odd and less than 5)

From given frequency distribution, we can see:

Sum is odd has following outcomes: ( 3, 5, 7, 9, 11) with corresponding frequencies ( 2, 4, 6, 4, 2)

Thus

P(Sum is odd) = (2+4+6+4+2) / 36

P(Sum is odd) = 18/36

and

Outcomes of Sum is less than 5 are ( 2, 3, 4) with frequencies ( 1 , 2 , 3 )

Thus

P( Sum is less than 5 ) = (1+2+3)/36

P( Sum is less than 5 ) = 6/36

and

From part a) we have:

P( Sum is odd and less than 5 ) = 2 / 36

Thus

P(Sum is odd or less than 5 )=P( Sum is Odd ) + P( Sum is less than 5) - P( Sum if odd and less than 5)

P(Sum is odd or less than 5 )= 18/36 + 6/36 - 2/36

P(Sum is odd or less than 5 )= (18+6-2) / 36

P(Sum is odd or less than 5 )= 22/36

P(Sum is odd or less than 5 )= 11 / 18