Chapter 10.5, Problem 79E

a.

To determine

To calculate:

Probability that at least $2$ people share same birthday for given events.

Answer to Problem 79E

Probability that at least $2$ people share same birthday for given events are $0.96$ and $0.88$ .

Explanation of Solution

Given information:

Group of $6$ randomly chosen people.

Group of $10$ randomly chosen people.

Calculation:

It is known that probability that at least $2$ people share the same birthday is $1$ minus no two people hare the same birthday.

In a group of randomly chosen people, the probability that six have different birthdays is as given,

  $\begin{array}{l}=\frac{365!}{\left(\left(365-n\right)!{.365}^n\right)}\\ =\frac{365!}{\left(\left(365-6\right)!{.365}^6\right)}\end{array}$

Require probability is,

  $P$ ( at least $2$ people share the same birthday),

  $=1-P$ (no two people share the same birthday)

  $\begin{array}{l}=1-\frac{365!}{\left(\left(365-6\right)!{.365}^6\right)}\\ =0.96\end{array}$

Thus, answer is $0.96$ .

In a group of $10$ randomly chosen people, the probability that six people have different birthdays is as given,

  $\begin{array}{l}=\frac{365!}{\left(\left(365-n\right)!{.365}^n\right)}\\ =\frac{365!}{\left(\left(365-10\right)!{.365}^{10}\right)}\end{array}$

Require probability is,

  $P$ ( at least $2$ people share the same birthday),

  $=1-P$ (no two people share the same birthday)

  $\begin{array}{l}=1-\frac{365!}{\left(\left(365-10\right)!{.365}^{10}\right)}\\ =0.88\end{array}$

Thus, answer is $0.88$ .

b.

To determine

To calculate:

Generalization the result from part $\left(a\right)$

Answer to Problem 79E

The formula can be generalized as below,

  $\frac{P{}_n}{365}$

Explanation of Solution

Given information:

Formula for $P\left(n\right)$ that at least $2$ people in a group of $n$ people share the same birthday.

Calculation:

The formula can be generalized as below,

  $\frac{P{}_n}{365}$

Where

  $P=$ Denotes permutations,

  $n=$ The size of the group.

c.

To determine

To calculate:

Group size for which probability that does at least $2$ people sharethe same birthday first exceed $50\%$ .

Answer to Problem 79E

Group size for which probability that does at least $2$ people sharethe same birthday first exceed $50\%$ is $23$ .

Explanation of Solution

Given information:

Formula for $P\left(n\right)$ that at least $2$ people in a group of $n$ people share the same birthday.

Calculation:

The result shows that for a group of $23$ people the probability of finding two people sharing some birthday exceed $50\%$ .

The probability is $0.5073$ .

Thus answer is $23$ .