Chapter 4, Problem 3CTC

To determine

To find: The number of people would need to be in a room so that there would be a 90% probability that2 people would be born on the same day.

To find: The number of people would need to be in a room so that there would be a 50% probability that 2 people would be born on the same day.

Answer to Problem 3CTC

The number of people would need to be in a room so that there would be a 90% probability that 2 people would be born on the same day is 39.

The number of people would need to be in a room so that there would be a 50% probability that 2 people would be born on the same day is 23.

Explanation of Solution

Given info:

Two people will have a same birthday and year is not a leap year.

Calculation:

Permutations Rule:

There are n different items and r number of selections and the selections are made without replacement, then the number of different permutations (order counts) is given by

${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

The formula for calculating the probability at least 2 people have the same birthday is,

$P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_k}{{365}^k}$

Here, k is the number of people

For $k=10$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{10}}{{365}^{10}}\\ =1-0.8831\\ =0.117\end{array}$

For $k=15$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{15}}{{365}^{15}}\\ =1-0.7471\\ =0.253\end{array}$

For $k=20$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{20}}{{365}^{20}}\\ =1-0.5886\\ =0.411\end{array}$

For $k=21$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{21}}{{365}^{21}}\\ =1-0.5563\\ =0.444\end{array}$

For $k=22$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{22}}{{365}^{22}}\\ =1-0.5243\\ =0.476\end{array}$

For $k=23$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{23}}{{365}^{23}}\\ =1-0.4927\\ =0.507\end{array}$

For $k=30$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{30}}{{365}^{30}}\\ =1-0.2937\\ =0.706\end{array}$

For $k=35$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{35}}{{365}^{35}}\\ =1-0.1856\\ =0.814\end{array}$

For $k=36$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{36}}{{365}^{36}}\\ =1-0.1678\\ =0.832\end{array}$

For $k=37$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{37}}{{365}^{37}}\\ =1-0.1513\\ =0.849\end{array}$

For $k=38$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{38}}{{365}^{38}}\\ =1-0.1359\\ =0.864\end{array}$

For $k=39$

$\begin{array}{c}P\left(\text{at}\text{least}\text{2}\text{people}\text{have}\text{same}\text{birthday}\right)=1-\frac{{}_{365}{P}_{39}}{{365}^{39}}\\ =1-0.1218\\ =0.878\end{array}$

It can be tabulated as follows:

Number of people	Probability that at least 2 have the same birthday
1	0.000
2	0.003
5	0.027
10	0.117
15	0.253
20	0.411
21	0.444
22	0.476
23	0.507
30	0.706
35	0.814
36	0.832
37	0.849
38	0.864
39	0.878

From the above table, it can be observed that the number of people would need to be in a room so that there would be a 90% probability that 2 people would be born on the same day is 39. The number of people would need to be in a room so that there would be a 50% probability that 2 people would be born on the same day is 23.