Textbook Problem

a. Prove that if A and B are any events in a sample space S, with $P\left(B\right)\ne 0,$ then $P\left(A^c|B\right)=1-P\left(A|B\right).$
b. Explain how the result in a part (a) justifies the following statements: (1) If the probability of a following statements: (1) If the probability of a false positive on a test for a condition is 4%, then there is a 96% probability that a person who does not have the condition will have a negative test result. (2) If the probability of a false negative on a test for a condition is 1%, then there is a 99% probability that a person who does have the condition will test positive for it.