Question

Consider the following experiment: we roll a fair die twice. The two rolls are independent events. Let’s call M the number of dots in the first roll and N the number of dots in the second roll.

(a) What is the probability that both M and N are even?

(b) What is the probability that M + N is even?

(c) What is the probability that M + N = 5?

(d) We know that M + N = 5. What is the probability that M is an odd number?

(e) We know that M is an odd number. What is the probability that M + N = 5?

Answer 1

(a) What is the probability that both M and N are even?
The total combinations on rolling a dice are given below.
(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)

Favourable event set is { (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) } = 9 possiblilities.

Hence the needed probability = 9/36 = 1/4

(b) What is the probability that M + N is even?

The figure below shows all possible sums on rolling the two dice.

From these those highlighted in blue are favorable event set, which is 18 possibliities.

Hence the needed probability = 18/36 = 1/2

(c) What is the probability that M + N = 5?
The figure below shows all possible sums on rolling the two dice.

From these those highlighted in blue are favorable event set, which is 4 possibliities.

Hence the needed probability = 4/36 = 1/9

(d) We know that M + N = 5. What is the probability that M is an odd number?

P(A and B) = P(A given B) P(B)

Event A is M+N =5
Event B is M = odd number

P(A given B) = 1/9 (Found the previous problem)
P(B) = 3/6 = 1/2 ( 3 numbers are odd out of 6 numbers on dice)

P(A and B) = 1/9 * 1/2 = 1/18

(e) We know that M is an odd number. What is the probability that M + N = 5?

P(A and B) = P(A given B) P(B)
Event A is M = odd number
Event B is M+N =5

P(A given B) = 3/6 = 1/2 ( 3 numbers are odd out of 6 numbers on dice)
P(B) = 4/36  

P(A and B) = 1/2 * 1/9 = 1/18