Consider an experiment with sample space = {1, 2, 3, 4, 5, 6, 7, 8}. The outcomes have probabilities

P[{1}] = 1 /4

P[{2}] = 1 /4

P[{3}] = 1 /8

P[{4}] = 1 /8

P[{5}] = 1 /16

P[{6}] = 1 /16

P[{7}] = 1/ 16

P[{8}] = 1 16 . We also define the events A = {1, 3, 4} B = {2, 3, 4} C = {3, 4, 5, 6, 7, 8} D = {2, 3, 5, 6} E = {2, 4, 6, 7} F = {5, 6, 7, 8} .

For each of the following questions, give a "Yes" or "No" answer as well as your reasoning and calculations.

(a) Are the events A, B, and C independent? If not, are they at least pairwise independent?

(b) Are the events A and D independent?

(c) Are the events A and F independent?

(d) Are the events B and independent?

(e) Are the events D, E, and F independent? If not, are they are least pairwise independent?

(f) Are the events A and D conditionally independent given C?