Consider the sample space S 1 2 3 20 and the events An

Consider the sample space S = { 1, 2, 3 ... 20 } and the events A_n= { x S | x is a multiple of n }, n = 1,2,3...

Which of the following pairs of events are independent?

(a) A2, A3

(b) A2, A4

Given S={1,2,3,…20}

A2={2,4,6,8,10,12,14,16,18,20}

A3={3,6,9,12,15,18}

A4={4,8,12,16,20}

a. A2 and A3 will be independent if p(A2 and A3)= p(A2).p(A3)

P(A2 and A3)= 3/20

P(A2)= 10/20 =1/2

P(A3)= 6/20

Rhs= 0.15 and Lhs= 0.15

Thus the pairs of events are independent.

A2 and A4 will be independent if p(A2 and A4)= p(A2).p(A4)

P(A2 and A4)=5/20 = 1/4

P(A2)= 10/20 = 1/2

P(A4)= 5/20= 1/4

Rhs= 0.125 and Lhs= 0.25

Thus the pairs of events are not independent.

c. A3 and A4 will be independent if p(A3 and A4)= p(A3).p(A4)

P(A3 and A4)=1/20 = 1/4

P(A3)= 6/20

P(A4)= 5/20= 1/4

Rhs= 0.075 and Lhs= 0.25

Thus the pairs of events are not independent.