Chapter 9.7, Problem 67E

a.

To determine

Find the probabilities that the $n$ people have distinct birthdays.

Answer to Problem 67E

The pattern gives the probabilities that $n$ people have distinct birthdays.

Explanation of Solution

Calculation:

Let us consider that a room is filled with $n$ number of people

If we consider $2$ people, that is $n=2$ then the probability that these two people will have distinct birthdays is

  $\begin{array}{l}=\frac{365}{365}\times \frac{364}{365}\\ =\frac{365\times 364}{{365}^2}\end{array}$

If we consider $3$ people, that is $n=3$ then the probability that these two people will have distinct birthdays is

  $\begin{array}{l}=\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\\ =\frac{363\times 364\times 365}{{365}^3}\end{array}$

Hence, the pattern gives the probabilities that $n$ people have distinct birthdays

b.

To determine

Write an expression for the probability.

Answer to Problem 67E

  $\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\times \frac{362}{365}$

Explanation of Solution

Calculation:

If we use the pattern in (a), then probability that $n=4$ people have distinct birthdays is

  $\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\times \frac{362}{365}$ .

c.

To determine

Verify that this probability can be obtained recursively.

Answer to Problem 67E

  $P_1=1$

Explanation of Solution

Calculation:

  $P_n$ is the probability that $n$ people have distinct birthdays

The probability that at least two people share the same birthday is the complement of the probability that no two people have the same birthday

Probability that $2$ people have distinct birthdays is $P_2=\frac{364}{365}$

Probability that $3$ people have distinct birthdays is

  $P_3=\frac{364}{365}\times \frac{363}{364}$

  $\frac{363}{365}$

Probability that $4$ people have distinct birthdays is

  $\begin{array}{l}{P}_4=\frac{364}{365}\times \frac{363}{364}\times \frac{362}{363}\\ =\frac{362}{365}\end{array}$

Hence probability that $n$ people have distinct birthdays is

  $P_n=1\times \left(1-\frac{1}{365}\right)\times \left(1-\frac{2}{365}\right)\times \mathrm{...}\times \left(1-\frac{n-1}{365}\right)$

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

If $n=1$ ,then

  $\begin{array}{l}{P}_1=\frac{365-(1-1)}{365}\\ P_1=\frac{365}{365}\end{array}$

  $P_1=1$

d.

To determine

Find the probability that at least two people in a group of $n$ people have the same birthday.

Answer to Problem 67E

  $Q_n=1-P_n$

Explanation of Solution

Calculation:

Let us try to find the probability that at least one person has the same birthday as you.

This will determine the probability that at least two people have the same birthday

An individual person will not have the same birthday as you with probability $P=\frac{364}{365}$

Hence the probability that n people all do not have your birthday is then $P_n$

So the probability that at least one person does have your birthday, or

Probability that at least two people have the same birthday is given by

  $Q_n=1-P_n$

e.

To determine

Complete the table.

Answer to Problem 67E

  $\begin{array}{cccccccc}n& 10& 15& 20& 23& 30& 40& 50\\ P_n& 0.88& 0.75& 0.59& 0.49& 0.29& 0.11& 0.03\\ Q_n& 0.12& 0.25& 0.45& 0.51& 0.71& 0.89& 0.97\end{array}$

Explanation of Solution

Calculation:

Let us put various values of $n$ and try to calculate $P_n$ and $Q_n$

Since $P_n=\left(\frac{365-(n-1)}{365}\right)\times P_{n-1}$ we will need to calculate all $P_n$ for $n$ from $1$ to $50$

We know that

  $p_1=1$

  $P_2=\left(\frac{365-(2-1)}{365}\right)\times P_{2-1}$

  $P_2=\frac{364}{365}$

  $P_9=0.905$

  $P_{10}=\left(\frac{365-(10-1)}{365}\right)\times P_{10-1}$

  $P_{10}=\left(\frac{365-9}{365}\right)\times P_9$

  $P_{10}=\left(\frac{356}{365}\right)\times 0.905$

  $P_{10}=0.88$

Similarly, we can calculate for all values for n from 1 to 50

Now, $Q_n=1-P_n$

We have already calculated requires values for $P_n$

We can easily calculate requires values for $Q_n$

Hence,

  $\begin{array}{l}{Q}_{10}=1-0.88\\ Q_{10}=0.12\end{array}$

Similarly, all required values can be calculated for different values of n from 1 to 50

Hence, we get the following answers after calculating for various values of n

  $\begin{array}{cccccccc}n& 10& 15& 20& 23& 30& 40& 50\\ P_n& 0.88& 0.75& 0.59& 0.49& 0.29& 0.11& 0.03\\ Q_n& 0.12& 0.25& 0.45& 0.51& 0.71& 0.89& 0.97\end{array}$

f.

To determine

Find the probability.

Answer to Problem 67E

  $23$

Explanation of Solution

Calculation:

Let us try to find how many people must be there in a group for a specific condition

From (d), we know that probability of at least two people having same birthday is given by $Q_n=1-P_n$

Also, from the table derives in (e) above, we see that for $n=23$

  $Q_n=0.51$

Hence, there must be $23$ in the group so that probability of at least two people having same birthday is greater than $\frac{1}{2}$