Emily is a songwriter who collects royalties on her songs whenever they are played in a commercial or a movie. Emily will earn $50 every time one of her songs is played in a commercial and she will earn $150 every time one of her songs is played in a movie. Emily's songs were played on 3 times as many commercials as movies and her total earnings on the royalties from all commercials and movies were $600. Graphically solve a system of equations in order to determine the number of commercials, x, x, and the number of movies, y, y, on which Emily's songs were played.

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.