[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 worker for the Department of Fish and Game is assigned

the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of them have tags.

• Without doing any fancy calculations, give a rough estimate of the number of trout in the lake.
• Let [math]N[/math] be the number of trout in the lake. Find an expression, in terms of [math]N[/math], for the probability that the worker would catch 10 tagged trout out of the 100 trout that she caught the second time.
• Find the value of [math]N[/math] which maximizes the expression in part (b). This value is called the maximum likelihood estimate for the unknown quantity [math]N[/math]. Hint: Consider the ratio of the expressions for successive values of [math]N[/math].