Chapter 6, Problem 70CE

In a certain Kentucky Fried Chicken franchise, half of the customers typically request “crispy” instead of “original.” (a) What is the probability that none of the next four customers will request “crispy”? (b) At least two? (c) At most two? (d) Construct the probability distribution (Excel or Appendix A), make a graph of its PDF, and describe its shape.

To determine

Find the probability that none of the customer will order light beer.

Answer to Problem 70CE

The probability for none of the customer will order light beer is 0.0168.

Explanation of Solution

Calculation:

The given information is that there is 40% of the U.S. beer drinkers order light beer.

The situation is a discrete probability distribution, in that the order light beer is success and does not order light beer is failure.

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 denotes the number of beer drinkers order light beer, the value of n is 8 and the value of $\pi$ is 0.40.

The probability for more than none of the customer will order light beer 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(0,8,0.40,0)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X=0\right)$ is 0.0168.

Thus, the probability for none of the customer will order light beer is 0.0168.

To determine

Find the probability one customer will order light beer.

Answer to Problem 70CE

The probability that one customer will order light beer is 0.0896.

Explanation of Solution

Calculation:

The probability for one customer will order light beer is, $P\left(X=1\right)$.

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(1,8,0.40,0)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X=1\right)$ is 0.0896.

Thus, the probability for one customer will order light beer is 0.0896.

To determine

Find the probability that two customers will order light beer.

Answer to Problem 70CE

The probability that two customers will order light beer is 0.2090.

Explanation of Solution

Calculation:

The probability for two customer will order light beer is, $P\left(X=2\right)$.

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(2,8,0.40,0)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X=2\right)$ is 0.2090.

Thus, the probability for two customers will order light beer is 0.2090.

To determine

Find the probability that fewer than three customers will order light beer.

Answer to Problem 70CE

The probability that fewer than three customers will order light beer is 0.3154.

Explanation of Solution

Calculation:

The probability for fewer than three customers will order light beer 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(2,8,0.40,1)”
• Output using EXCEL software is given below:

From the output, the value of $P\left(X\le 2\right)$ is 0.3154.

Thus, the probability for fewer than three customers will order light beer is 0.3154.

To determine

Construct the probability distribution.

Draw the graph of PDF and find the shape of the distribution.

Answer to Problem 70CE

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, $\text{mean }<\text{median}<\mathrm{mode}$.

Construct the probability distribution and graph:

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 8 and probability occurrence as 0.40.
• Select Display graph.
• Click on Ok.
• Output using EXCEL software is given below:

Observation:

From the output, the tail of the distribution elongated slightly towards to right.

Thus, the distribution shape is right skewed.