Product rule which involves Bayes Theorem P(X,Y)=P(X)P(Y∣X)=P(Y)P(X∣Y)=P(X∩Y)P(X, Y) = P(X)P(Y|X) = P(Y)P(X|Y) = P(X \cap Y) P(X,Y)=P(X)P(Y∣X)=P(Y)P(X∣Y)=P(X∩Y)Joint probability of X and Y is simply understood as choosing X first and choose Y given X. Confusing part is that regardless of order, reverse probability is expressed as same form.After that we can decide Pairwise Independence or notP(X)=P(X∣Y),P(Y)=P(Y∣X)P(X) = P(X|Y), P(Y) = P(Y|X)P(X)=P(X∣Y),P(Y)=P(Y∣X)P(z∣x,y)P(x,y)=P(x,y,z)=P(x,y∣z)P(z)P(z|x,y) P(x, y) = P(x, y, z) = P(x,y|z) P(z)P(z∣x,y)P(x,y)=P(x,y,z)=P(x,y∣z)P(z)Bonferroni inequalityP(X∩Y)≥P(A)+P(B)−1P(X \cap Y) \ge P(A) + P(B) - 1P(X∩Y)≥P(A)+P(B)−1
