We flip 3 coins with all 8 outcomes equally likely: S = {(H, H, H), . . . ,(T, T, T)}. Define
the following random variables
X = 1, if the first and second flip are both Heads
0, else
Y = 1, if the first and third flip are both Heads
0, else
Define events A = {X = 1}, B = {Y = 1}, C = {first flip is Heads}
a) Find the PMFs of X and Y.
b) Compute P[A], P[B], P[A ∩ B]. Are A and B independent?
c) Compute P[A|C], P[B|C], P[A ∩ B|C]. Are A and B conditionally independent given C?
Note: Events A and B are conditionally independent given event C, if P(A∩B|C) = P(A|C)P(B|C)