Chapter 8.7, Problem 67E

(a)

To determine

The justification for the pattern $n=2:\frac{365}{365}\cdot \frac{364}{365}=\frac{365\cdot 364}{{365}^2}$, $n=3:\frac{365}{365}\cdot \frac{364}{365}\cdot \frac{363}{365}=\frac{365\cdot 364\cdot 363}{{365}^3}$ for the probabilities of $n$ people having distinct birthdays in a group.

(b)

To determine

The pattern for $n=4$ people similar to the pattern $n=2:\frac{365}{365}\cdot \frac{364}{365}=\frac{365\cdot 364}{{365}^2}$, $n=3:\frac{365}{365}\cdot \frac{364}{365}\cdot \frac{363}{365}=\frac{365\cdot 364\cdot 363}{{365}^3}$ for the probabilities of $n$ people having distinct birthdays in a group.

(c)

To determine

To prove: The pattern for the probability of $n$ people having distinct birthdays in a group, $P_n$ is recursively obtained by $P_1=1\text{ and }{P}_n=\frac{365-\left(n-1\right)}{365}{P}_{n-1}$.

(d)

To determine

The justification about the formula $Q_n=1-P_n$ which gives the probability that at least two persons in a group of $n$ people have the same birth date.

(e)

To determine

To fill: The provided table by using the pattern for the probability of $n$ people having distinct birthdays in a group, $P_n$ which is obtained by the formula $P_1=1\text{ and }{P}_n=\frac{365-\left(n-1\right)}{365}{P}_{n-1}$ and the formula $Q_n=1-P_n$ gives the probability that at least two persons in a group of $n$ people having the same birth date; where the provided table is:

$\begin{array}{rrrrrrrr}\hfill n& \hfill 10& \hfill 15& \hfill 20& \hfill 23& \hfill 30& \hfill 40& \hfill 50\\ \hfill P_n& \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill Q_n& \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

(f)

To determine

The number of people in a group for which the probability of at least two people having the same birthday is greater than $\frac{1}{2}$.