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 $3.$

$(20$ 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 between $1$ 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 $)$ 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 $)$ function.