Chapter 6.4, Problem 23SE

In the Ardmore Hotel, 20 percent of the customers pay by American Express credit card. (a) Of the next 10 customers, what is the probability that none pay by American Express? (b) At least two? (c) Fewer than three? (d) What is the expected number who pay by American Express? (e) Find the standard deviation. (f) Construct the probability distribution (using Excel or Appendix A). (g) Make a graph of its PDF, and describe its shape.

To determine

Find the probability that no one will pay by American Express.

Answer to Problem 23SE

The probability for none of them pay by American express out of 10 customers is 0.1074.

Explanation of Solution

Calculation:

The given information is that in Ardmore hotel, the customers pay by American express credit card is nearly 20%.

Binomial Distribution:

The random variable X denotes the number of successes in trail of n.

The PDF of the distribution is,

$P\left(X=x\right)=\frac{n!}{x!\left(n-x\right)!}{\pi }^x{\left(1-\pi \right)}^{n-x}$

Here n denotes the number of trails and $\pi$ denotes the value of the probability for success.

In the given situation, the random variable X variable denotes the number of customers, the value of n is 10 and the value of $\pi$ is 0.20.

The probability for no one pay by American express card in next 10 customers is $P\left(X=0\right)$.

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL software is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=BINOM.DIST(0,10,0.20,0)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X=0\right)$ is 0.1074.

Thus, the probability for none of them pay by American express out of 10 customers is 0.1074.

To determine

Find the probability that at least two customers will pay by American Express.

Answer to Problem 23SE

The probability that at least two customers will pay by American Express is 0.6242.

Explanation of Solution

Calculation:

The probability for at least two customers pay by American express card in next 10 customers is,

$P\left(X\ge 2\right)=1-P\left(X\le 1\right)$.

Probability = 1– BINOM.DIST(x, n, probability, cumulative)

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL software is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=BINOM.DIST(1,10,0.20,1)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X\le 1\right)$ is 0.3758.

Then the value of $P\left(X\ge 2\right)$ is,

$\begin{array}{c}P\left(X\ge 2\right)=1-0.3758\\ =0.6242\end{array}$

Thus, the probability for at least 2 customers will pay by American express out of 10 customers is 0.6242.

To determine

Find the probability that fewer than 3 customers will pay by American Express.

Answer to Problem 23SE

The probability that fewer than 3 customers will pay by American Express is 0.6778.

Explanation of Solution

Calculation:

The probability for fewer than 3 customers pay by American express card in next 10 customers is,

$P\left(X<3\right)=P\left(X\le 2\right)$.

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL software is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=BINOM.DIST(2,10,0.20,1)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X<3\right)$ is 0.6778.

Thus, the probability for fewer than 3 customers will pay by American express out of 10 customers is 0.6778.

To determine

Find the expected number of customers will by American express.

Answer to Problem 23SE

The expected number of customers will by American express is 2.

Explanation of Solution

Calculation:

The formula for expected value is,

$\mu =n\pi$

The expected number of customers is,

$\begin{array}{c}\mu =n\pi \\ =10\times 0.20\\ =2\end{array}$

Thus, the expected number of customers pay by American express is 2.

To determine

Find the standard deviation.

Answer to Problem 23SE

The standard deviation is 1.2649.

Explanation of Solution

Calculation:

The formula for standard deviation is,

$\sigma =\sqrt{n\pi \left(1-\pi \right)}$

Calculate the standard deviation is,

$\begin{array}{c}\sigma =\sqrt{10\times 0.20\left(1-0.20\right)}\\ =\sqrt{10\times 0.20\left(0.80\right)}\\ =\sqrt{1.6}\\ =1.2649\end{array}$

Thus, the standard deviation is 1.2649.

To determine

Construct the probability distribution.

Explanation of Solution

Calculation:

Construct the probability distribution:

Software procedure:

Step by step procedure to obtain probability distribution by using MegaStat software is as follows:

• Open an EXCEL file, Go to Add-ins, select MegaStat.
• Choose Probability, select the Discrete probability distributions.
• In Binomial, enter number of trails as 10 and probability occurrence as 0.2.
• Click on Ok.
• Output using EXCEL software is given below:

To determine

Draw the graph for PDF and shape of the distribution.

Answer to Problem 23SE

The distribution shape is right-skewed.

Explanation of Solution

Calculation:

Shape of the distribution:

Symmetric distribution:

When the left and right sides of the distribution are approximately equal or mirror images of each other, then it is symmetric distribution. For this distribution, mean and standard deviation are useful. For a symmetric distribution, $\text{mean}=\text{median}=\text{mode}$.

Right skewed distribution:

If the tail of the distribution is elongated towards the right, then it is a right-skewed distribution. It is also called as positively skewed. For right skewed, $\text{mean}>\text{median}>\mathrm{mode}$.

Left-skewed distribution:

If the tail of the distribution is elongated towards the left, then it is a left-skewed distribution. It is also called as negatively skewed. For left skewed, $\text{mean }<\text{median}<\mathrm{mode}$.

Graph of the probability distribution:

Software procedure:

Step by step procedure to obtain probability distribution by using MegaStat software is as follows:

• Open an EXCEL file, Go to Add-ins, select MegaStat.
• Choose Probability, select the Discrete probability distributions.
• In Binomial, enter number of trails as 10 and probability occurrence as 0.2.
• Select Display graph.
• Click on Ok.
• Output using EXCEL software is given below:

Observation:

From the output, the tail of the distribution is elongated towards the right.

Thus, the distribution shape is right-skewed.