Consider a liquid of volume V, which contains N bacteria. Let the liquid be vigorously shaken and part of it transferred to a test tube of volume v. Suppose that (i) the probability p that any given bacterium will be transferred to the test tube is equal to the ratio of the volumes vi V and that (ii) the appearance of I particular bacterium in the test tube is independent of the appearance of the other N - I bacteria. Consequently, the number of bacteria in the test tube is a numerical valued random phenomenon obeying a binomial probability law with parameters N and p = v/V. Let m = N/V denote the average number of bacteria per unit volume. Let the volume v of the test tube be equal to 3 cubic centimeters.
(i) Assume that the volume v of the test tube is very small compared to the volume V of liquid, so that p = v/V is a small number. In particular, assume that p = 0.001 and that the bacterial density m = 2 bacteria per cubic centimeter. Find approximately the probability that the number of bacteria in the test tube will be greater than I.
(ii) Assume that the volume v of the test tube is comparable to the volume V of the liquid. In particular, assume that V = 12 cubic centimeters and N = 10,000. What is the probability that the number of bacteria in the test tube will be between 2400 and 2600, inclusive?