Consider an amnesic miner trapped in a room that contains three doors. Door 1 leads him to freedom after two-day’s travel; door 2 returns him to his room after four-day’s travel; and door 3 returns...

Consider an amnesic miner trapped in a room that contains three doors. Door 1 leads him to freedom after two-day’s travel; door 2 returns him to his room after four-day’s travel; and door 3 returns him to his room after eight-day’s travel. Suppose each door is equally likely to be chosen whenever he is in the room, and let T denote the time it takes the miner to become free.

a) Define a sequence of independent and identically distributed random variables X₁, X₂, . . . and a stopping rule J such that