13. Consider a miner trapped in a room which contains three doors. Door 1 leads him to freedom after two-days' travel; door 2 returns him to his room after four-days' journey; and door 3 returns him to his room after sixdays'

journey. Suppose at all times he is equally likely to choose any of the three doors, and let Ô denote the time it takes the miner to become free.

(a) Define a sequence of independent and identically distributed random variables Xx, X2,... and a stopping time Í such that

Í

ô= Ó ÷, i'= 1 Note: You may have to imagine that the miner continues to randomly choose doors even after he reaches safety.

(b) Use Wald's equation to find E[T].

(c) Compute Å[Ó?=éXt\N = n] and note that it is not equal to

ÅßÓÀ-é×,é

(d) Use part

(c) for a second derivation of E[T].