2 rhinos are charging towards each other from a distance of 100 metres apart. They are travelling at 5 metres per second. A fly starts from the horn of one of the rhinos and flies directly to the horn of the other rhino at a speed of 12 metres per second. When it arrives it immediately turns and flies back to the horn of the other rhino and repeats this behaviour until the 2 rhinos collide.

How far does the fly travel during this time?
An efficient, elegant method is required to solve this problem

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.