Calculate the probability P$($n$)$ that n spins are up out of a total of N for N $=4$ and N $=16$ and put your results in a table. Calculate the mean values of n and n$^2$ using your tabulated values of PN$($n$).$ It is possible to do the calculation for general p and q$,$ but choose p $=$ q $=1/2$ for simplicity. Although it is better to first do the calculation of PN $($n$)$ by hand, you can use Program Binomial.

$($b$)$ Use Program Binomial to plot Py $($n$)$ for larger values of N$.$ Assume that p $=$ q $=1/2.$ Deter$-$ mine the value of n corresponding to the maximum of the probability and visually estimate the width for each value of N$.$ What is your measure of the width? One measure is the value of n at which Py$($n$)$ is equal to half its value at its maximum. What is the qualitative dependence of the width on N$?$ Also compare the relative heights of the maximum of Py for increasing values of N$.$

$($c$)$ Program Binomial also plots P$($n$)$ versus n$/.$ Does the width of Py $($n$)$ appear to become larger or smaller as N is increased?

$($d$)$ Plot in Py $($n$)$ versus n for N $=16.($Choose Log Axes under the views menu.$)$ Describe the qualitative dependence of in Py $($n$)$ on n$.$ Can in Py$($n$)$ be fitted to a paraboln of the form A$+$B$(-),$ where A and B are fit parameters