Exercise $7\mathrm{.}27-$ The Birthday Problem

Make a function BirthdayProblem.m with the declaration

function $p=$ BirthdayProblem$(n)$

where n is a number of people and p is a probability vector. You can find the details of the Birthday Problem in the

textbook, but I am making the following modifications to the instructions:

The Birthday Problem primary function will create an $(n-1)-$vector of probabilities and a plot of those probabilities versus

n$.$ You will perform a numerical experiment $($described in the problem's hint$)$ in a subfunction that your primary

function calls.

So if I input

$p=$ BirthdayProblem$(40)$

p should be a vector of length $n-1$ that holds the probabilities of $2,3,4,$dots,$40$ people sharing a birthday and your

program should create a plot of these probabilities. Inside your program, each of these individual probabilities should

be calculated by calling a subfunction that runs the numerical experiment for a given value of people. Use a loop call

the subfunction to do the experiment $n-1$ times.

$7\mathrm{.}27$ The Birthday Problem The Birthday Problem is as follows: If there are a group of $n$

people in a room, what is the probability that two or more of them have the same

birthday? It is possible to determine the answer to this question by simulation. Write a

function that calculates the probability that two or more of $n$ people will have the same

birthday, where $n$ is a calling argument. $($Hint: To do this, the function should create an

array of size $n$ and generate $n$ birthdays in the range $1$ to $365$ randomly. It should then

check to see if any of the $n$ birthdays are identical. The function should perform this

experiment at least $5000$ times and calculate the fraction of those times in which two or

more people had the same birthday.$)$ Write a test program that calculates and prints out

the probability that $2$ or more of $n$ people will have the same birthday for $n=2,3,$dots,$40.$