Four friends are trying to decide what kind of cake to order for a party: cheesecake, chocolate, or yellow. everyone was asked to vote for their favorite, and the winner under the unknown voting method was chocolate. upon hearing the results, two people who had voted for yellow cake wanted to change their votes to move chocolate to the top of their preference lists. the winner was recalculated, and cheesecake was the winner which fairness criterion has been violated in this election?

Write the equation in logarithmic form 8^x=64

Interpret the following expression (5y^2-7)^3/(2y)^2

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.