The us forest service requires a report of deer population in us national parks. teams use a method called "capture-recapture" to estimate wildlife populations. to estimate the population over a wide area, samples are taken in which a number of deer are captured, tagged, and released back into the environment to mingle with the rest of the herd. later, a larger sample of deer are captured, some of which are tagged and some are not. parks officials use the ratio of tagged to untagged deer in the sample to project the population for the entire herd. nick and his team initially collected, tagged, and released 24 deer. several days later, the teams returned to the area and captured 55 deer, of which 9 were tagged. find the estimated number of deer in this population (to the nearest whole number) if we presume that this sample ratio is typical for the entire herd.

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.