[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

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 [math]N[/math] denote the actual number of people who live there. Assume that the census counted [math]n_1[/math] people living in this area. Now, another census was taken in the locality, and [math]n_2[/math] people were counted. In addition, [math]n_{12}[/math] people were counted both times.

• Given [math]N[/math], [math]n_1[/math], and [math]n_2[/math], let [math]X[/math] denote the number of people counted both times. Find the probability that [math]X = k[/math], where [math]k[/math] is a fixed positive integer between 0 and [math]n_2[/math].
• Now assume that [math]X = n_{12}[/math]. Find the value of [math]N[/math] which maximizes the expression in part (a). Hint: Consider the ratio of the expressions for successive values of [math]N[/math].