$*1\mathrm{.}20$ Project: The scientific method. The scientific method has four elements: Observe, Theorize, Predict, Test. For example, observations and careful thought lead to the theory of dice sums presented in Section $1.$ In this project we will derive predictions from that theory and test them by means of a simulation. Follow the instructions, and present your results in a one$-$page technical report.

$($a$)$ Using the theory presented at the end of Section $1,$ give a table of expected frequencies ${?}^2$ of all possible sums, in the experiment of tossing a pair of fair dice $10,000$ times. Round all expected frequencies to the nearest integer.

$($b$)$ Use R to carry out a simulation of $10000$ tosses of two dice. Give the observed frequencies of all sums. Display these frequencies by adding an additional line to the table of expected frequencies from part $($a$).$ To do the simulation, activate $R$ and do the following.

Set the random number seed to $26**Y+K$ where $Y$ denotes the last digit of the current year and $K$ is the position in the alphabet of the first letter of your last name. If your last name begins with $C,$ and the year is $2023,$ then your random number seed is $81,$ and so you would enter the command set. seed$(81).$

Create a variable, say ac$,$ and set it equal to NULL by the command ac NULL.

Run a program like the following the one$-$line program:

for $($i in $1$:$10000)\{$w sample $,$ replace$=$TRUE$)$;