A census in the United States is an attempt to count everyone in the country. It is inevitable that many people are not counted. The U. S. Census Bureau proposed a way to estimate the number of people who were not counted by the latest census. Their proposal was as follows: In a given locality, let N denote the actual number of people who live there. Assume that the census counted n1 people living in this area. Now, another census was taken in the locality, and n2 people were counted. In addition, n12 people were counted both times.
(a) Given N, n1, and n2, let X denote the number of people counted both times. Find the probability that X=k, where k is a fixed positive integer between 0 and n2.
(b) Now assume that X=n12. Find the value of N which maximizes the expression in part (a). Hint: Consider the ratio of the expressions for successive values of N.