solution
Say Omega = N\{0}, sigma algebra is power set of N and the probability mass function is p(n)=2^-n .
Then A is all the even numbers, B all numbers at least 4, C all numbers at least 5.
P(A)=sum{2^-n^ | n is even} = sum{2^-2n^ | n is at least 1} = sum{4^-n^ | n is at least 1} = 1/(1-1/4)-1=1/3
P(B) = P(N\{1,2,3}) = 1 - 1/2 - 1/4 - 1/8 = 1/8
P(C) = 1/16 similarly
P(A and B) = P(A\{2}) = 1/3 - 1/4 = 1/12 =/= P(A)P(B) therefore not independent
P(A and C) = P(A\{2,4}) = P(A)P(C) with a similar calculation and therefore independent