Problem of the Week: Semi-Sanitary Autumnal Fun
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
THE RULES:
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Send solutions to bkennedy@gettysburg.edu. This problem was posted on Saturday, November 6 and solutions are due on Friday, November 12 by 5:00 p.m.
THE PROBLEM:
10 identical apples are in a barrel at a party. One guest will randomly choose two apples from the barrel, and then randomly choose one of these two apples to return to the barrel; there are now 9 apples in the barrel. Another guest will randomly choose two apples from the barrel and return one, leaving 8 apples in the barrel. Suppose that eight guests in total complete this process; there are now only two apples in the barrel.
THE QUESTION:
At this point, what is the probability that one of the apples remaining in the barrel has never yet been chosen? Answer, and explain. Can you generalize to the situation where there are initially n apples in the barrel?
