How large does need to be in other words how many time do

Question

Suppose we wish to estimate the probability,P_A, of some event, A. We do so by repeating an experiment times and observing whether or not the event occurs during each experiment. In particular, let

(a) Assuming n is large enough so that the central limit theorem applies, find an expression for 

(b) Suppose we want to be 95% certain that our estimate is within  of the true value. That is, we want   How large does need to be? In other words, how many time do we need to run the experiment?

(c) Let be the number of times that we observe the event during our repetitions of the experiment. That is  Assuming that is chosen according to the results of part (b), find an expression for the average number of times the event is observed  Show that for rare events  is essential independent of P_A . Thus, even if we have no idea about the true value of P_A , we can run the experiment until we observe the event for a predetermined number of times and be assured of a certain degree of accuracy in our estimate of pA.