# Find the probability that the rat dies by poison.

Mathematical statistics- Statistical inference

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Problem 3. Consider the maze shown below. There are three cells (Cell 1, Cell 2, and Cell 3) and two

deadly (quite permanent) outcomes (Death By Poison, and the dreaded Death By Guillotine). A rat is

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Order Paper Nowinitially placed in cell 1. When the rat enters Cell i, he wanders around within the cell for Xi minutes,

where Xi is uniformly distributed between 0 and i, and then he exits the cell by picking one of the doors at

random (e.g., if there are 3 doors, he picks each with probability 1/3).

(a) Find the probability that the rat dies by poison. (Recall that he starts in Cell 1.)

(b) What is the expected number of minutes that the rat lives?

(c) What is the probability the rat visits Cell 3 before he dies?

(In each of the above problems, you should DEFINE a set of unknowns precisely, set up a system of equations

involving the unknowns, and then solve for the nal answer (time permitting). You need not solve the actual

system, if you give a complete description of how you would nish solving the problem.)