Question

Two ordinary fair, six-sided dice ate rolled.
What is the probability the sum of the numbers on the two dice is 6, given that the number on at least one of the dice is 3?
What is the probability the sum of the numbers on the two dice is 8, given that the number on at least one of the dice is 3?
What is the probability that the sum of the numbers on the two dice is 9, given that it is not 3?
What is the probability that exactly one of the dice shows the number 1 given that the sum of the number is 3?

Answer 1

Probability = No. of desired outcomes / total outcomes

The event is rolling a pair of dice (2 dice). So the sample space of one observation is

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) }

n = 36 outcomes

Sum of the outcomes can range from 2 - 12

The following are the eg: of outcomes

2 = {(1,1)} n(2) = 1

3 = { (1,2) , (2,1)} n(3) = 2

7 = { (1,6), (2,5) (3,4), (4,3), (5,2) , (6,1) } n(7) = 6

We can have the pmf then

x (sum)	No. of outcomes (1)
2	1
3	2
4	3
5	4
6	5
7	6
8	5
9	4
10	3
11	2
12	1
Total	36

What is the probability the sum of the numbers on the two dice is 6, given that the number on at least one of the dice is 3?

his is conditional probability where we know that one event has happened but on its happening what is the possibility of that event happening in a particular way. We know that the sum is 6 but it happening in a way that the at least 1 roll is 3 (1 roll is three or both rolls can be 3)

Probability = (Rolling at least 1 three) / Getting 6

Event of getting 1 three and sum 6 = { (3,3) } n = 1

Probability ..............n(6) can be found from the table

What is the probability the sum of the numbers on the two dice is 8, given that the number on at least one of the dice is 3?

Probability = (Rolling at least 1 three) / Getting 8

Rolling at least 1 three = { (3,5) , (5,3) } n = 2 P = 2/36

Probability =   ..............n(8) can be found from the table

What is the probability that the sum of the numbers on the two dice is 9, given that it is not 3?

Probabiltiy = Not getting any 3 / Sum 9

Not getting 3 = { (4,5) , (5,4) } n = 2

Probabiltiy =   ..............n(9) can be found from the table

  

What is the probability that exactly one of the dice shows the number 1 given that the sum of the number is 3?

Sum = 3

the possbilities with = { (1,2) , (2,1) } n = 2

Probability = 2 / 2   ..............n(3) can be found from the table

To get a sum 3 we will indefinitedly have '1' as one of the outcome. It is a certain event