A turtle challenges the Greek warrior Achilles to a race outside the gates of Troy. Achilles accepts. The turtle and Achilles start from the gates of Troy and race in a straight line away from the city. They keep a
constant speed throughout the race. During the race, Achilles is f meters from the gates, while the turtle is
only g meters from the gates.
Which of the following expressions always represents the distance between the turtle and Achilles during
the race?
Choose 1

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.