6.1.4 Consider an experiment in which a certain event will occur with probability h and will not occur with probability 1????h, where is a fixed positive parameter and h is a small .h / positive variable. Suppose that n independent trials of the experiment are carried out, and the total number of times that the event occurs is noted. Show that

(a) The probability that the event never occurs during the n trials is 1????nhC o.h/;

(b) The probability that the event occurs exactly once is nhCo.h/;

(c) The probability that the event occurs twice or more is o.h/.

Hint: Use the binomial expansion

(1-ah)" 1-nah+ n(n-1) (ah) -.... 2