91Ó°ÊÓ

(a) we suppose we flip $50$coins, $n=50$. The total number of microstates is:

$2^n=2^{50}=1.1259×{10}^{15}\text{microstate}$

(b) The number of microstates receiving $25$heads is as follows:

$\left(\begin{array}{l}N\\ n\end{array}\right)$

where n denotes the number of heads and N denotes the number of coins, so:

$\left(\begin{array}{l}50\\ 25\end{array}\right)=\frac{50!}{25!(50−25)!}=1.2641×{10}^{14}\text{microstate}$

(c)Regardless of order, the probability of getting the macrostate of heads and tails is:

$P=\frac{\left(\begin{array}{l}N\\ n\end{array}\right)}{\text{Total number of microstates}}$

$→P=\frac{\left(\begin{array}{l}50\\ 25\end{array}\right)}{2^{50}}=0.1123$

(d) The probability of getting the macrostate of$30$heads and $20$tails, regardless of order, is:

$→P=\frac{\left(\begin{array}{l}50\\ 30\end{array}\right)}{2^{50}}=\frac{\frac{50!}{30!(50−30)!}}{2^{50}}=0.04186$

(e)The probability of obtaining the macrostate of$40$heads and$10$tails, regardless of order, is as follows:

localid="1650287550141" $→P=\frac{\left(\begin{array}{c}50\\ 40\end{array}\right)}{2^{50}}=\frac{\frac{50!}{40!(50−40)!}}{2^{50}}=9.1236×{10}^{−6}$

(f) Regardless of order, the probability of receiving the macrostate of$50$heads and $0$tails is:

localid="1650287553981" $→P=\frac{\left(\begin{array}{c}50\\ 0\end{array}\right)}{2^{50}}=\frac{\frac{50!}{0!(50−0)!}}{2^{50}}=8.882×{10}^{−16}$

(g)The function we need to plot is the number of heads divided by the probability, so is:

localid="1650287557923" $P\left(n\right)=\frac{\mathrm{Î\copyright }(50,n)}{2^{50}}=\frac{\frac{50!}{n!(50−n)!}}{2^{50}}$

The table shows the values that we intend to plot where n varies from$0$to$50$. I drew the graph in MS Excel, and the image below shows how I generated the data.

The probability decreases sharply as we move away from the midpoint, with an equal number of heads and tails. This is nicely illustrated by a plot of the probability P(N):