There is a looming crisis in universities and colleges across north america. in most places, enrollments are increasing requiring more instructors. however, there are not enough phds to fill the vacancies now. moreover, among current professors, a large proportion are nearing retirement age. on top of these problems, some universities allow professors over the age of 60 to retire early. to devise a plan to deal with the crisis, a consultant surveyed 521 55-to 64-year-old professors and asked each whether he or she intended to retire before 65. the responses are 1 = no and 2 = yes. estimate with 95% confidence the proportion of professors who plan on early retirement.

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.