Question

Two fair six sided dice are rolled.

(i) What is the probability that the smaller value of the two results is 3 given that the sum of the two results is 8?

(ii) What is the probability that the sum of the two results is at most 5 given that the number 2 appeared at least once?

(iii) What is the probability that the sum of the two results is 7 given that exactly one of the two results is odd?

Answer 1

Total event space when two dice are rolle and sum of the two results;

Dice1	Dice 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

(i)

Event space when sum of the two results is '8' along with smaller of the two results

Dice1	Dice 2	Sum	smaller value of two results
2	6	8	2
3	5	8	3
4	4	8	4
5	3	8	3
6	2	8	2

Total number of event that sum of the two result is '8' = 5

Out of these '5' events, number of events that the smaller value is '3' =2

Probability that the smaller value of the two results is 3 given that the sum of the two results is 8 = 2/5

(ii)

Following is the event set whne the number apperead atleast once along with Sum atmost 5(yes), Sum more than 5(no)

Dice1	Dice 2	Sum	Sum atmost 5
2	1	3	Yes
2	2	4	Yes
2	3	5	Yes
2	4	6	No
2	5	7	No
2	6	8	No

1	2	3	Yes
3	2	5	Yes
4	2	6	No
5	2	7	No
6	2	8	No

Total number of event which result in the number 2 appeared at least once =11

Out of these event number of events where the sum is atmost 5 (5) = 5

probability that the sum of the two results is at most 5 given that the number 2 appeared at least once = 5/11

(iii) probability that the sum of the two results is 7 given that exactly one of the two results is odd

Event space when one of the two results is off long with the sum of the two results

Event	Dice1	Dice 2	Sum
1	1	2	3
2	1	4	5
3	1	6	7
4	3	2	5
5	3	4	7
6	3	6	9
7	5	2	7
8	5	4	9
9	5	6	11
10	2	1	3
11	2	3	5
12	2	5	7
13	4	1	5
14	4	3	7
15	4	5	9
16	6	1	7
17	6	3	9
18	6	5	11

Number of events that exactly one of the two results is odd = 18

Out these events number of events where the sum of the two results is 7 = 6

Probability that the sum of the two results is 7 given that exactly one of the two results is odd = 6/18 = 1/3