Consider the following variation of the final round of the classic TV game show $$Let$$s

Make A Deal$.$ There are three doors, and behind one of them there is a car, while behind

the other two are goats. You are asked to choose one of the doors, and, without loss of

generality, you choose Door $1.$ Now, the host Monty Hall opens either Door $2$ or Door

$3,$ behind which is a goat. $($He never opens up the door with the car behind it$.)$ Monty

now gives you the choice: do you want to stick with Door $1,$ or switch to the other door.

What should you do$?$ Or does it matter?

In this problem we will create a Monte Carlo simulation of the generalized form of the

Monty Hall problem. Instead of three doors, we will have n doors, where n can be any

number of doors $>=3.$

$1.(20$ pts$)$ The Monty Hall problem requires a good deal of generating random numbers.

We will begin by creating and testing two helper functions to generate random numbers

for us$.$

NOTE: For full credit be sure to use the rand function in the generation of your random

numbers.

$($a$)(5$ pts$)$ Write a Matlab function f $=$ randnum$($a$,$b$)$ which takes as input two integers

a and b$,$ and generates and returns a random integer that is equally likely to take any

value between a and b $($inclusive$).$ Save this function in the file randnum.m$.$

$($b$)(5$ pts$)$ Write a Matlab m$-$file test$\_$randnum.m that tests the accuracy of your randnum$()$

function. This program should use randnum$()$ to generate $100,000$ random numbers be$-$

tween $1$ and $10.$ Store each random number in a vector x $($it will be of length $100,000$ when

the simulation is done$).$ Then display a histogram of the results using the hist$($x$,1$:$10)$

function.

$($c$)(5$ pts$)$ Write a Matlab function f $=$ randnum$\_$reject$($a$,$b$,$c$)$ which takes as input

three integers a$,$ b and c$,$ and generates and returns a random integer that is equally likely

to take any value between a and b $($inclusive$),$ but will not return the value given by c$.$

Save this function in the file randnum$\_$reject.m$.$

$($d$)(5$ pts$)$ Write a Matlab m$-$file test$\_$randnum$\_$reject.m that tests the accuracy of your

randnum$\_$reject$()$ function. This program should use randnum$\_$reject$()$ to generate

$100,000$ random numbers between $11$ and $20$ excluding $15.$ Store each random number

in a vector x $($it will be of length $100,000$ when the simulation is done$).$ Then display a

histogram of the results using the hist$($x$,11$:$20)$ function.

$1$