The diagram below indicates how probabilities have been assigned to various subsets of the sample space \(S\) :

a. Are the three events \(A, B\), and \(C\) pairwise independent events?

b. Are the three events \(A, B\), and \(C\) independent events?

c. What is the value of \(P(A \cap B)\) ? What is the value of \(P(A \cap B \mid C)\) ?

d. Suppose event \(D\) is such that \(P(D)=.05\) and \(D \cap(A \cup B \cup C)=\emptyset\). Are events \(D\) and \(A\) independent? Are events \(D\) and \((A \cup B \cup C)\) independent?

e. What is the probability of event \(C\), given \((A \cap B)\) ?

A .15 .05 S .10 .10 .175 .05 .15 B C