Chapter 9.7, Problem 67E

(a)..

To determine

To find: The way in which the given pattern gives the possibility that the $n$ people have distinct birthdays.

Answer to Problem 67E

The given statement is False.

Explanation of Solution

Given:

The given pattern is,

  $\begin{array}{c}n=2:\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\\ =\frac{365\cdot 364}{{365}^2}\end{array}$

Also,

  $\begin{array}{c}n=3:\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right)\\ =\frac{365\cdot 364\cdot 363}{{365}^3}\end{array}$

Calculation:

Consider for $n=2$ , the number of people that will have two distinct birthdays is,

  $\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)=\frac{365\cdot 364}{{365}^2}$

Consider for $n=3$ , the number of people that will have two distinct birthdays is,

  $\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right)=\frac{365\cdot 364\cdot 363}{{365}^3}$

Thus, the two pattern shows that the $n$ people have distinct birthdays.

(b)..

To determine

To find: The probability that $n=4$ people have distinct birthdays.

Answer to Problem 67E

The probability is $\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right)\left(\frac{362}{365}\right)$ .

Explanation of Solution

Calculation:

Consider the given pattern is,

  $\begin{array}{c}n=2:\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\\ =\frac{365\cdot 364}{{365}^2}\end{array}$

Also,

  $\begin{array}{c}n=3:\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right)\\ =\frac{365\cdot 364\cdot 363}{{365}^3}\end{array}$

Consider the pattern for $n=4$ , the people have distinct birthday as,

  $\left(\frac{365}{365}\right)\left(\frac{364}{365}\right)\left(\frac{363}{365}\right)\left(\frac{362}{365}\right)$

(c)..

To determine

To prove: The probability for the n number of people is $P_n$ and it is obtained recursively.

Explanation of Solution

Given:

The recursive equation to determine the probability is,

  $P_n=\frac{365-\left(n-1\right)}{365}{P}_{n-1}$

Calculation:

Consider the probability that 2 people have distinct birthday is,

  $P_2=\frac{364}{365}$

Consider the probability that 3 people have distinct birthday is,

  $P_2=\frac{364}{365}\cdot \frac{363}{364}$

Consider the probability that 4 people have distinct birthday is,

  $\begin{array}{c}{P}_2=\frac{364}{365}\cdot \frac{363}{364}\cdot \frac{362}{363}\\ =\frac{362}{365}\end{array}$

Thus, the probability for the n people is,

  $\begin{array}{c}{P}_n=1\left(1-\frac{1}{365}\right)\left(1-\frac{2}{365}\right)\left(1-\frac{n-1}{365}\right)\\ =\left(\frac{365-\left(n-1\right)}{365}\right)P_{n-1}\end{array}$

Hence, proved.

(d)..

To determine

To find: The way in which the given expression gives the probability that at least two people in a group of n people have the same birthday.

Answer to Problem 67E

The probability that at least two person have the birthday is $Q_n=1-P_n$ .

Explanation of Solution

Given:

The given equation is,

  $Q_n=1-P_n$

Calculation:

Consider the person with the probability of birthday $P=\frac{364}{365}$ will not have the same probability as the other person.

Thus, the probability that at least two person have the birthday is,

  $Q_n=1-P_n$

(e)..

To determine

To find: The completed form of the given table.

Answer to Problem 67E

The completed table is shown in Table 2..

Explanation of Solution

Given:

The given table is shown in Table 1

Table 1

$n$	10	15	20	23	30	40	50
$P_n$
$Q_n$

Calculation:

Consider the formula for the probability of n people is,

  $P_n=\left(\frac{365-\left(n-1\right)}{365}\right)P_{n-1}$

Consider the formula to calculate the probability that at least two people have same birthday is,

  $Q_n=1-P_n$

Then, the completed table is shown in Table 2

Table 2

$n$	10	15	20	23	30	40	50
$P_n$	0.88	0.75	0.59	0.49	0.29	0.11	0.03
$Q_n$	0.12	0.25	0.45	0.51	0.71	0.89	0.97

(f)..

To determine

To find: The number of people that must be there so that the probability of at least two of the man having same birthday is more than $\frac{1}{2}$ .

Answer to Problem 67E

The number of people in the group must be 23.

Explanation of Solution

Consider the formula to calculate the probability that at least two people have same birthday is,

  $Q_n=1-P_n$

From the table 2.

  $Q_{23}=0.51$

SO, the number of people in the group must be 23.