Answer the following questions regarding the validity of probability assignments. If you answer false, explain why the statement is false.

a. If \(P(A)=.2, P(B)=.3\), and \(A \cap B=\emptyset\), then \(P(A \cup B)\) \(=.06\). True or False?

b. If \(A \cap B=\emptyset\) and \(P(B)=.2\), then \(P(A \mid B)=0\). True or False?

c. If \(P(B)=.05, P(A \mid B)=.80\), and \(P(A \mid \bar{B})=.5\), then \(P(B \mid A)=.0777\) (to four digits of accuracy). True or False?

d. If \(P(A)=.8\) and \(P(B)=.7\), then \(P(A \cap B) \geq .5\). True or False?

e. It is possible that \(P(A)=.7, P(B)=.4\), and \(A \cap B=\emptyset\). True or False?