Josh wanted to buy a bicycle but didn't have enough money. after josh told his troubles to sam slick, sam said,"i can fix that. see that fence? each time you jump that fence, i'll double your money. there's one small thing though: you must give me $32 each time for the priviledge of jumping." josh agreed, jumped the fence, received his payment from sam slick, and paid sam $32. repeating the routine twice more, josh was distressed to find that, on the last jump, after sam had made his payment to josh, josh had only $32, with which to pay sam and so had nothing left. sam, of course, went merrily on his way, leaving josh wishing that he had known a little more about mathematics.

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.