Chapter 5.4, Problem 31P

a.

To determine

Explain the reason for a negative binomial distribution to be appropriate for a given random variable and define a formula for $P\left(n\right)$ in the given context of application.

Answer to Problem 31P

There are binomial trials with probability of success 0.80 and failure 0.20 with n as a random variable that represents a number of contacts needed to get 12^th successful sale.

The formula for probability that Person S makes 12 successful sales is. $\underset{\_}{P\left(n\right)=C_{n-1,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^{n-12}}$

Explanation of Solution

Calculation:

There are binomial trials with probability of success $p=0.80$ and failure as $q=0.20$.

Let n follow a negative binomial distribution that represents the number of contacts needed to get 12^th sale.

Probability of success $p=0.80$

Probability of failure $q=0.20$

Number of successful sales $k=12$

Negative binomial probability:

The probability that k^th success occurs on n^th trial is given below:

$P\left(n\right)=C_{n-1,k-1}{p}^k{q}^{n-k}$

Here, n is the number of trials in which k^th  success occurs, k is the number of successes, $p$ is the probability of success, and $q$ is the probability of failure.

The formula for probability that Person S makes 12 successful sales is given below:

$\begin{array}{l}P\left(n\right)=C_{n-1,k-1}{p}^k{q}^{n-k}\\ =C_{n-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{n-12}\\ =C_{n-1,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^{n-12}\end{array}$

Thus, the formula for probability that Person S makes 12 successful sales is $\underset{\_}{P\left(n\right)=C_{n-1,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^{n-12}}$.

b.

To determine

Calculate the given probabilities.

Answer to Problem 31P

The probability that Person S needs 12 contacts to get bonus is 0.0687.

The probability that Person S needs 13 contacts to get bonus is 0.1649.

The probability that Person S needs 14 contacts to get bonus is 0.2144.

Explanation of Solution

Calculation:

The probability that Person S needs 12 contacts to get bonus is given below:

$\begin{array}{l}P\left(12\right)=C_{12-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{12-12}\\ =C_{11,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^0\\ =\left(\frac{11!}{11!0!}\right){\left(0.80\right)}^{12}{\left(0.20\right)}^0\\ =0.0687\end{array}$

Thus, the probability that Person S needs 12 contacts to get bonus is 0.0687.

The probability that Person S needs 13 contacts to get bonus is given below:

$\begin{array}{l}P\left(13\right)=C_{13-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{13-12}\\ =C_{12,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^1\\ =\left(\frac{12!}{11!1!}\right){\left(0.80\right)}^{12}{\left(0.20\right)}^1\\ =0.1649\end{array}$

Thus, the probability that Person S needs 13 contacts to get bonus is 0.1649.

The probability that Person S needs 14 contacts to get bonus is given below:

$\begin{array}{l}P\left(14\right)=C_{14-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{14-12}\\ =C_{13,11}{\left(0.80\right)}^{12}{\left(0.20\right)}^2\\ =\left(\frac{13!}{11!2!}\right){\left(0.80\right)}^{12}{\left(0.20\right)}^2\\ =0.2144\end{array}$

Thus, the probability that Person S needs 14 contacts to get bonus is 0.2144.

c.

To determine

Calculate the probability that Person S will require 12 to 14 contacts to get bonus.

Answer to Problem 31P

The probability that Person S will require 12 to 14 contacts to get bonus is 0.4480.

Explanation of Solution

Calculation:

The probability that Person S will require 12 to 14 contacts to get bonus is calculated as follows:

$\begin{array}{l}P\left(12\le n\le 14\right)=P\left(12\right)+P\left(13\right)+P\left(14\right)\\ =C_{12-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{12-12}+C_{13-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{13-12}+C_{14-1,12-1}{\left(0.80\right)}^{12}{\left(0.20\right)}^{14-12}\\ =0.0687+0.1649+0.2144\\ =\text{0}\text{.4480}\end{array}$

Thus, the probability that Person S will require 12 to 14 contacts to get bonus is 0.4480.

d.

To determine

Calculate the probability that Person S will require more than 14 contacts to get bonus.

Answer to Problem 31P

The probability that Person S will require more than 14 contacts to get bonus is 0.5520.

Explanation of Solution

Calculation:

The probability that Person S will require more than 14 contacts to get bonus is calculated follows:

$\begin{array}{l}P\left(n>14\right)=1-P\left(n\le 14\right)\\ =1-P\left(12\le n\le 14\right)\\ =1-\text{0}\text{.4480}\\ =\text{0}\text{.5520}\end{array}$

Thus, the probability that Person S will require more than 14 contacts to get bonus is 0.5520.

e.

To determine

Calculate the expected value of $n$.

Calculate the standard deviation of $n$.

Answer to Problem 31P

The expected value of $n$.is 15.

The standard deviation of $n$ is 1.94.

Explanation of Solution

Calculation:

The expected value of $n$ is calculated as follows:

$\begin{array}{l}\mu =\frac{k}{p}\\ =\frac{12}{0.80}\\ =15\end{array}$

Thus, the expected value of $n$ is 15.

The standard deviation of $n$ is calculated as follows:

$\begin{array}{l}\sigma =\frac{\sqrt{kq}}{p}\\ =\frac{\sqrt{12\times 0.20}}{0.80}\\ =1.94\end{array}$

Thus, the standard deviation of $n$ is 1.94.

Interpretation:

The expected number of contacts that the twelfth sale will occur is 15 with a standard deviation of 1.94.