Probability questions for discrete mathematics.

Question

1. Let X = {2, 3, 4}. Suppose that m is chosen uniformly at random from X. Then n is chosen uniformly at random from X. Let G be the random variable whose value is gcd(m, n). Let A be the event that G = 2. Let B be the event that m = n.

(i) What is the distribution of G?

(ii) What is the expected value of G?

(iii) What is the variance of G?

(iv) What is Pr(A)?

(v) What is Pr(A | B)?

(vi) Are A and B independent events?

Answer 1

P(m = 2) = 1/3

P(m = 3) = 1/3

P(m = 4) = 1/3

P(n = 2) = 1/3

P(n = 3) = 1/3

P(n = 4) = 1/3

G = gcd (m,n)

Combinations (m, n) = (2,2) (2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4)

--------------- gcd (m,n) = 2 -------1------2------1------3-----1------2------1-----4

9 values, 4 x 1, 3 x 2, 1 x 3, 1 x 4

3 values where m=n 2,2 3,3 4,4

hence

P(G=1) = 4/9

P(G=2) = 3/9 = 1/3 = P(A)

P(G=3) = 1/9

P(G=4) = 1/9

P(B) = P(m=n) = 3/9 = 1/3

... etc