Chapter 15.3, Problem 7P

(a) Note that (3.4) assumes $P(A)\ne 0$ since $P_A(B)$ is meaningless if $P(A)=0$ Assuming both $P(A)\ne 0$ and $P(B)\ne 0,$ show that if (3.4) is true, then $P(A)=P_B(A);$ that is if $B$ is independent of A, then $A$ is independent of $B$ If either $P(A)$ or $P(B)$ is zero, then we use (3.5) to define independence.

(b) When is an event $E$ independent of itself? When is $E$ independent of "not $E^{\prime \prime }?$