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Applied Statistics in Business and Economics

David Doane, Lori Seward

Chapter 6

Discrete Probability Distributions - all with Video Answers

Educators

Chapter Questions

02:52

Problem 1

Which of the following could not be probability distributions? Explain
$$
\begin{array}{lr}
{\text { Example } \mathbf{A}} \\
\hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{x}) \\
\hline 0 & .80 \\
1 & .20
\end{array}
$$
$$
\begin{aligned}
&\text { Example B }\\
&\begin{array}{cc}
\hline \boldsymbol{x} & \boldsymbol{P ( \boldsymbol { x } )} \\
\hline 1 & .05 \\
2 & .15 \\
3 & .25 \\
4 & .40 \\
5 & .10 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{array}{cc}
{\text { Example C }} \\
\hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{x}) \\
\hline 50 & .30 \\
60 & .60 \\
70 & .40
\end{array}
$$

Willis James
Willis James
Numerade Educator
07:42

Problem 2

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day. (a) Find $P(X=75)$. (b) Find $P(X \leq 75)$. (c) Find $P(X>50)$. (d) Find $P(X<100)$. (e) Which of the probability expressions in parts (a)-(d) is a value of the CDF?
$$
\begin{array}{cccccc}
\hline \boldsymbol{x} & 25 & 50 & 75 & 100 & \text { Total } \\
\boldsymbol{P ( \boldsymbol { x } )} & .20 & .40 & .30 & .10 & 1.00 \\
\hline
\end{array}
$$

Robin Corrigan
Robin Corrigan
Numerade Educator
02:49

Problem 3

On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below. (a) Find $P(X \geq 3)$. (b) Find $P(X \leq 2)$. (c) Find $P(X<4)$. (d) Find $P(X=1)$. (e) Which of the probability expressions in parts (a)-(d) is a value of the CDF?
$$
\begin{array}{cccccccc}
\hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & \text { Total } \\
\boldsymbol{P}(\boldsymbol{x}) & .05 & .30 & .25 & .20 & .15 & .05 & 1.00 \\
\hline
\end{array}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:59

Problem 4

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day. (a) Calculate the expected value and standard deviation of this random variable $X$ by using the PDF shown. (b) Describe the shape of this distribution.
$$
\begin{array}{cccccc}
\hline \boldsymbol{x} & 25 & 50 & 75 & 100 & \text { Total } \\
\boldsymbol{P}(\boldsymbol{x}) & .20 & .40 & .30 & .10 & 1.00 \\
\hline
\end{array}
$$

Joshua Argo
Joshua Argo
Numerade Educator
05:33

Problem 5

On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below. (a) Calculate the mean and standard deviation. (b) Describe the shape of this distribution.
$$
\begin{array}{clllllll}
\hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & \text { Total } \\
\boldsymbol{P}(\boldsymbol{x}) & .05 & .30 & .25 & .20 & .15 & .05 & 1.00 \\
\hline
\end{array}
$$

Zachary Zhao
Zachary Zhao
Numerade Educator
04:09

Problem 6

Pepsi and Mountain Dew products sponsored a contest giving away a Lamborghini sports car worth $$\$ 215,000$$. The probability of winning from a single bottle purchase was .00000884 . Find the expected value. Show your calculations clearly. (Data are from J. Paul Peter and Jerry C. Olson, Consumer Behavior and Marketing Strategy, 7th ed. [McGraw-Hill/Irwin, 2005], p. 226.)

Ernest Castorena
Ernest Castorena
Numerade Educator
01:23

Problem 7

Student Life Insurance Company wants to offer an insurance plan with a maximum claim amount of $$\$ 5,000$$ for dorm students to cover theft of certain items. Past experience suggests that the probability of a maximum claim is .01. What premium should be charged if the company wants to make a profit of $$\$ 25$$ per policy? Assume any student who files a claim files for the maximum amount and there is no deductible. Show your calculations clearly.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:43

Problem 8

A lottery ticket has a grand prize of $$\$ 28$$ million. The probability of winning the grand prize is .000000023 . Based on the expected value of the lottery ticket, would you pay $$\$ 1$$ for a ticket? Show your calculations and reasoning clearly.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:52

Problem 9

Oxnard Petro Ltd. is buying hurricane insurance for its off-coast oil drilling platform. During the next five years, the probability of total loss of only the above-water superstructure ( $$\$ 250$$ million) is .30, the probability of total loss of the facility ($$\$950$$ million) is .30 , and the probability of no loss is .40. Find the expected loss.

Bailey Brooks
Bailey Brooks
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03:50

Problem 10

Find the mean and standard deviation of four-digit uniformly distributed lottery numbers $(0000$ through 9999).

Sophie Knight
Sophie Knight
Numerade Educator
02:19

Problem 11

The ages of Java programmers at SynFlex Corp. range from 20 to 60. (a) If their ages are uniformly distributed, what would be the mean and standard deviation? (b) What is the probability that a randomly selected programmer's age is at least 40 ? At least 30 ? Hint: Treat employee ages as integers.

Richard Miller
Richard Miller
Numerade Educator
02:08

Problem 12

Use Excel to generate 100 random integers from (a) 1 through 2, inclusive; (b) 1 through 5 , inclusive; and (c) 0 through 99, inclusive. (d) In each case, write the Excel formula. (e) In each case, calculate the mean and standard deviation of the sample of 100 integers you generated, and compare them with their theoretical values.

Lucas Finney
Lucas Finney
Numerade Educator
03:12

Problem 13

List the $X$ values that are included in each italicized event.
a. You can miss at most 2 quizzes out of 16 quizzes ( $X=$ number of missed quizzes).
b. You go to Starbuck's at least 4 days a week ( $X=$ number of Starbuck's visits).
c. You are penalized if you have more than 3 absences out of 10 lectures ( $X=$ number of absences).

Asma Venkitta
Asma Venkitta
Numerade Educator
04:38

Problem 14

Write the probability of each italicized event in symbols (e.g., $P(X \geq 5)$.
a. At least 7 correct answers on a 10-question quiz ( $X=$ number of correct answers).
b. Fewer than 4 "phishing" e-mails out of 20 e-mails ( $X=$ number of phishing e-mails).
c. At most 2 no-shows at a party where 15 guests were invited ( $X=$ number of no-shows).

Chris Wojturski
Chris Wojturski
Numerade Educator
04:25

Problem 15

Find the mean and standard deviation for each binomial random variable:
a. $n=8, \pi=.10$
b. $n=10, \pi=.40$
c. $n=12, \pi=.50$

Danielle Flores
Danielle Flores
Numerade Educator
04:25

Problem 16

Find the mean and standard deviation for each binomial random variable:
a. $n=30, \pi=.90$
b. $n=80, \pi=.70$
c. $n=20, \pi=.80$

Danielle Flores
Danielle Flores
Numerade Educator
01:30

Problem 17

Calculate each binomial probability:
a. $X=5, n=9, \pi=.90$
b. $X=0, n=6, \pi=.20$
c. $X=9, n=9, \pi=.80$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:26

Problem 18

Calculate each binomial probability:
a. $X=2, n=8, \pi=.10$
b. $X=1, n=10, \pi=.40$
c. $X=3, n=12, \pi=.70$

Robin Corrigan
Robin Corrigan
Numerade Educator
02:04

Problem 19

Calculate each compound event probability:
a. $X \leq 3, n=8, \pi=.20$
b. $X>7, n=10, \pi=.50$
c. $X<3, n=6, \pi=.70$

Jon Southam
Jon Southam
Numerade Educator
02:00

Problem 20

Calculate each compound event probability:
a. $X \leq 10, n=14, \pi=.95$
b. $X>2, n=5, \pi=.45$
c. $X \leq 1, n=10, \pi=.15$

Jon Southam
Jon Southam
Numerade Educator
02:42

Problem 21

Calculate each binomial probability:
a. More than 10 successes in 16 trials with an 80 percent chance of success.
b. At least 4 successes in 8 trials with a 40 percent chance of success.
c. No more than 2 successes in 6 trials with a 20 percent chance of success.

Nick Johnson
Nick Johnson
Numerade Educator
02:35

Problem 22

Calculate each binomial probability:
a. Fewer than 4 successes in 12 trials with a 10 percent chance of success.
b. At least 3 successes in 7 trials with a 40 percent chance of success.
c. At most 9 successes in 14 trials with a 60 percent chance of success.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
03:57

Problem 23

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.

Robin Corrigan
Robin Corrigan
Numerade Educator
04:41

Problem 24

Historically, 5 percent of a mail-order firm's repeat charge-account customers have an incorrect current address in the firm's computer database. (a) What is the probability that none of the next 12 repeat customers who call will have an incorrect address? (b) One customer? (c) Two customers?
(d) Fewer than three? (e) Construct the probability distribution (using Excel or Appendix A), make a graph of its PDF, and describe its shape.

A M
A M
Numerade Educator
04:59

Problem 25

At a Noodles \& Company restaurant, the probability that a customer will order a nonalcoholic beverage is .38. Use Excel to find the probability that in a sample of 5 customers (a) none of the 5 will order a nonalcoholic beverage, (b) at least 2 will, (c) fewer than 4 will, (d) all 5 will order a nonalcoholic beverage.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:59

Problem 26

J.D. Power and Associates says that 60 percent of car buyers now use the Internet for research and price comparisons. (a) Find the probability that in a sample of 8 car buyers, all 8 will use the Internet; (b) at least 5; (c) more than 4. (d) Find the mean and standard deviation of the probability distribution. (e) Sketch the PDF (using Excel or Appendix A) and describe its appearance (e.g., skewness). (Data are from J. Paul Peter and Jerry C. Olson, Consumer Behavior and Marketing Strategy, 7th ed. [McGraw-Hill/Irwin, 2005], p. 188.)

Nick Johnson
Nick Johnson
Numerade Educator
04:39

Problem 27

There is a 70 percent chance that an airline passenger will check bags. In the next 16 passengers that check in for their flight at Denver International Airport, find the probability that (a) all will check bags; (b) fewer than 10 will check bags; (c) at least 10 will check bags.

Cathy Wang
Cathy Wang
Numerade Educator
01:38

Problem 28

Police records in the town of Saratoga show that 15 percent of the drivers stopped for speeding have invalid licenses. If 12 drivers are stopped for speeding, find the probability that (a) none will have an invalid license; (b) exactly one will have an invalid license; (c) at least 2 will have invalid licenses.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 29

Find the mean and standard deviation for each Poisson:
a. $\lambda=1.0$
b. $\lambda=2.0$
c. $\lambda=4.0$

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04:14

Problem 30

Find the mean and standard deviation for each Poisson:
a. $\lambda=9.0$
b. $\lambda=12.0$
c. $\lambda=7.0$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:52

Problem 31

Calculate each Poisson probability:
a. $P(X=6), \lambda=4.0$
b. $P(X=10), \lambda=12.0$
c. $P(X=4), \lambda=7.0$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:09

Problem 32

Calculate each Poisson probability:
a. $P(X=2), \lambda=0.1$
b. $P(X=1), \lambda=2.2$
c. $P(X=3), \lambda=1.6$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:53

Problem 33

Calculate each compound event probability:
a. $P(X \leq 3), \lambda=4.3$
b. $P(X>7), \lambda=5.2$
c. $P(X<3), \lambda=2.7$

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
02:53

Problem 34

Calculate each compound event probability:
a. $P(X \leq 10), \lambda=11.0$
b. $P(X>3), \lambda=5.2$
c. $P(X<2), \lambda=3.7$

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:25

Problem 35

Calculate each Poisson probability:
a. More than 10 arrivals with $\lambda=8.0$.
b. No more than 5 arrivals with $\lambda=4.0$.
c. At least 2 arrivals with $\lambda=5.0$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:14

Problem 36

Calculate each Poisson probability:
a. Fewer than 4 arrivals with $\lambda=5.8$.
b. At least 3 arrivals with $\lambda=4.8$.
c. At most 9 arrivals with $\lambda=7.0$.

Christopher Stanley
Christopher Stanley
Numerade Educator

Problem 37

According to J.D. Power and Associates' 2006 Initial Quality Study, consumers reported on average 1.7 problems per vehicle with new 2006 Volkswagens. In a randomly selected new Volkswagen, find the probability of (a) at least one problem; (b) no problems; (c) more than three problems. (d) Construct the probability distribution using Excel or Appendix B, make a graph of its PDF, and describe its shape. (Data are from J.D. Power and Associates 2006 Initial Quality Study ${ }^{\mathrm{SM}}$.)

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02:47

Problem 38

At an outpatient mental health clinic, appointment cancellations occur at a mean rate of 1.5 per day on a typical Wednesday. Let $X$ be the number of cancellations on a particular Wednesday. (a) Justify the use of the Poisson model. (b) What is the probability that no cancellations will occur on a particular Wednesday? (c) One? (d) More than two? (e) Five or more?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
09:57

Problem 39

The average number of items (such as a drink or dessert) ordered by a Noodles \& Company customer in addition to the meal is 1.4. These items are called add-ons. Define $X$ to be the number of add-ons ordered by a randomly selected customer. (a) Justify the use of the Poisson model. (b) What is the probability that a randomly selected customer orders at least 2 add-ons? (c) No add-ons? (d) Construct the probability distribution using Excel or Appendix B, make a graph of its PDF, and describe its shape.

Chris Trentman
Chris Trentman
Numerade Educator
03:32

Problem 40

(a) Why might the number of yawns per minute by students in a warm classroom not be a Poisson event? (b) Give two additional examples of events per unit of time that might violate the assumptions of the Poisson model, and explain why.

Mahnoor Khan
Mahnoor Khan
Numerade Educator
07:31

Problem 41

An experienced order taker at the L.L. Bean call center has a .003 chance of error on each keystroke (i.e., $\pi=.003$ ). In 500 keystrokes, find the approximate probability of (a) at least two errors and (b) fewer than four errors. (c) Is the Poisson approximation justified?

Robin Corrigan
Robin Corrigan
Numerade Educator
13:50

Problem 42

The probability of a manufacturing defect in an aluminum beverage can is .00002 . If 100,000 cans are produced, find the approximate probability of (a) at least one defective can and (b) two or more defective cans. (c) Is the Poisson approximation justified?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:13

Problem 43

Three percent of the letters placed in a certain postal drop box have incorrect postage. Suppose 200 letters are mailed. (a) For this binomial, what is the expected number with incorrect postage? (b) For this binomial, what is the standard deviation? (c) What is the approximate probability that at least 10 letters will have incorrect postage? (d) Fewer than five? (e) Is the Poisson approximation justified?

Raj Bala
Raj Bala
Numerade Educator
01:36

Problem 44

In a string of 100 Christmas lights, there is a .01 chance that a given bulb will fail within the first year of use (if one bulb fails, it does not affect the others). Find the approximate probability that two or more bulbs will fail within the first year.

Narayan Hari
Narayan Hari
Numerade Educator
01:39

Problem 45

The probability that a passenger's bag will be mishandled on a U.S. airline is .0046. During spring break, suppose that 500 students fly from Minnesota to various southern destinations. (a) What is the expected number of mishandled bags? (b) What is the approximate probability of no mishandled bags? More than two? (c) Would you expect the approximation to be accurate (cite a rule of thumb)?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:22

Problem 46

(a) State the values that $X$ can assume in each hypergeometric scenario. (b) Use the hypergeometric PDF formula to find the probability requested. (c) Check your answer by using Excel.
$$
\begin{aligned}
\text { i. } N & =10, n=3, s=4, P(X=3) \\
\text { ii. } N & =20, n=5, s=3, P(X=2) \\
\text { iii. } N & =36, n=4, s=9, P(X=1) \\
\text { iv. } N & =50, n=7, s=10, P(X=3)
\end{aligned}
$$

Nick Johnson
Nick Johnson
Numerade Educator
04:46

Problem 47

ABC Warehouse has eight refrigerators in stock. Two are side-by-side models and six are topfreezer models. (a) Using Excel, calculate the entire hypergeometric probability distribution for the number of top-freezer models in a sample of four refrigerators chosen at random. (b) Make an Excel graph of the PDF for this probability distribution and describe its appearance.

Abdul Vahid M
Abdul Vahid M
Numerade Educator
02:36

Problem 48

A statistics textbook chapter contains 60 exercises, 6 of which are essay questions. A student is assigned 10 problems. Define $X$ to be the number of essay questions the student receives. (a) Use Excel to calculate the entire hypergeometric probability distribution for $X$. (b) What is the probability that none of the questions are essay? (c) That at least one is essay? (d) That two or more are essay? (e) Make an Excel graph of the PDF of the hypergeometric distribution and describe its appearance.

Dilip Paruchuri
Dilip Paruchuri
Numerade Educator
03:26

Problem 49

Fifty employee travel expense reimbursement vouchers were filed last quarter in the finance department at Ramjac Corporation. Of these, 20 contained errors. A corporate auditor inspects five vouchers at random. Let $X$ be the number of incorrect vouchers in the sample. (a) Use Excel to calculate the entire hypergeometric probability distribution. (b) Find $P(X=0)$. (c) Find $P(X=$ 1). (d) Find $P(X \geq 3)$. (e) Make an Excel graph of the PDF of the hypergeometric distribution and describe its appearance.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:00

Problem 50

A medical laboratory receives 40 blood specimens to check for HIV. Eight actually contain HIV. A worker is accidentally exposed to five specimens. (a) Use Excel to calculate the entire hypergeometric probability distribution. (b) What is the probability that none contained HIV? (c) Fewer than three? (d) At least two? (e) Make an Excel graph of the PDF of the hypergeometric distribution and describe its appearance.

Sajin Shajee
Sajin Shajee
Numerade Educator
02:56

Problem 51

(a) Check whether the binomial approximation is acceptable in each of the following hypergeometric situations. (b) Find the binomial approximation (using Appendix A) for each probability requested. (c) Check the accuracy of your approximation by using Excel to find the actual hypergeometric probability.
a. $N=100, n=3, s=40, P(X=3)$
b. $N=200, n=10, s=60, P(X=2)$
c. $N=160, n=12, s=16, P(X=1)$
d. $N=500, n=7, s=350, P(X=5)$

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:52

Problem 52

Two hundred employee travel expense reimbursement vouchers were filed last year in the finance department at Ramjac Corporation. Of these, 20 contained errors. A corporate auditor audits a sample of five vouchers. Let $X$ be the number of incorrect vouchers in the sample. (a) Justify the use of the binomial approximation. (b) Find the probability that the sample contains no erroneous vouchers. (c) Find the probability that the sample contains at least two erroneous vouchers.

Christopher Stanley
Christopher Stanley
Numerade Educator
20:56

Problem 53

A law enforcement agency processes 500 background checks for firearms purchasers. Fifty applicants are convicted felons. Through a clerk's error, 10 applicants are approved without checking for felony convictions. (a) Justify the use of the binomial approximation. (b) What is the probability that none of the 10 is a felon? (c) That at least 2 of the 10 are convicted felons? (d) That fewer than 4 of the 10 are convicted felons?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:06

Problem 54

Four hundred automobiles are to be inspected for California emissions compliance. Of these, 320 actually are compliant but 80 are not. A random sample of 6 cars is chosen. (a) Justify the use of the binomial approximation. (b) What is the probability that all are compliant? (c) At least 4 ?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:53

Problem 55

Find each geometric probability.
a. $P(X=5)$ when $\pi=.50$
b. $P(X=3)$ when $\pi=.25$
c. $P(X=4)$ when $\pi=.60$

Robin Corrigan
Robin Corrigan
Numerade Educator
01:02

Problem 56

In the Ardmore Hotel, 20 percent of the guests (the historical percentage) pay by American Express credit card. (a) What is the expected number of guests until the next one pays by American Express credit card? (b) What is the probability that the first guest to use an American Express is within the first 10 to checkout?

Christopher Stanley
Christopher Stanley
Numerade Educator

Problem 57

In a certain Kentucky Fried Chicken franchise, half of the customers request "crispy" instead of "original," on average. (a) What is the expected number of customers before the next customer requests "crispy"? (b) What is the probability of serving more than 10 customers before the first request for "crispy"?

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Problem 58

The height of a Los Angeles Lakers basketball player averages 6 feet 7.6 inches (i.e., 79.6 inches) with a standard deviation of 3.24 inches. To convert from inches to centimeters, we multiply by 2.54. (a) In centimeters, what is the mean? (b) In centimeters, what is the standard deviation? (c) Which rules did you use?

Ariana Nash
Ariana Nash
Numerade Educator
03:59

Problem 59

July sales for Melodic Kortholt, Ltd., average $$\mu_1=\$ 9,500$$ with $$\sigma_1{ }^2=\$ 1,250$$. August sales average $$\mu_2=\$ 7,400$$ with $$\sigma_2{ }^2=\$ 1,425$$. September sales average $$\mu_3=\$ 8,600$$ with $$\sigma_3{ }^2=\$ 1,610$$. (a) Find the mean and standard deviation of total sales for the third quarter. (b) What assumptions are you making?

Manisha Sarker
Manisha Sarker
Numerade Educator
05:04

Problem 60

The mean January temperature in Fort Collins, CO, is $37.1^{\circ} \mathrm{F}$ with a standard deviation of $10.3^{\circ} \mathrm{F}$. Express these Fahrenheit parameters in degrees Celsius using the transformation $\mathrm{C}=5 / 9 \mathrm{~F}-17.78$.

Arulmozhi T
Arulmozhi T
Numerade Educator
09:19

Problem 61

There are five accounting exams. Bob's typical score on each exam is a random variable with a mean of 80 and a standard deviation of 5. His final grade is based on the sum of his exam scores. (a) Find the mean and standard deviation of Bob's point total assuming his performances on exams are independent of each other. (b) By the Empirical Rule (see Chapter 4), would you expect that Bob would earn at least 450 points (the required total for an "A")?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:35

Problem 62

The probability that a 30 -year-old white male will live another year is .99863 . What premium would an insurance company charge to break even on a one-year $$\$ 1$$ million term life insurance policy?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:08

Problem 63

As a birthday gift, you are mailing a new personal digital assistant (PDA) to your cousin in Toledo. The PDA cost $$\$ 250$$. There is a 2 percent chance it will be lost or damaged in the mail. Is it worth $$\$ 4$$ to insure the mailing? Explain, using the concept of expected value.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:11

Problem 64

Use Excel to generate 1,000 random integers in the range 1 through 5. (a) What are the expected mean and standard deviation? (b) What are your sample mean and standard deviation? (c) Is your sample consistent with the uniform model? Discuss. (d) Show the Excel formula you used.

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
01:03

Problem 65

Consider the Bernoulli model. What would be a typical probability of success $(\pi)$ for (a) free throw shooting by a good college basketball player? (b) Hits by a good baseball batter? (c) Passes completed by a good college football quarterback? (d) Incorrect answers on a five-part multiple choice exam if you are guessing? (e) Can you suggest reasons why independent events might not be assumed in some of these situations? Explain.

Dominador Tan
Dominador Tan
Numerade Educator
02:53

Problem 66

There is a 14 percent chance that a Noodles \& Company customer will order bread with the meal. Use Excel to find the probability that in a sample of 10 customers (a) more than five will order bread; (b) no more than two will; (c) none of the 10 will order bread. (d) Is the distribution skewed left or right?

Anand Jangid
Anand Jangid
Numerade Educator
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Problem 67

In a certain year, on average 10 percent of the vehicles tested for emissions failed the test. Suppose that five vehicles are tested. (a) What is the probability that all pass? (b) All but one pass? (c) Sketch the probability distribution and discuss its shape.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:28

Problem 68

The probability that an American CEO can transact business in a foreign language is .20. Ten American CEOs are chosen at random. (a) What is the probability that none can transact business in a foreign language? (b) That at least two can? (c) That all 10 can?

AG
Ankit Gupta
Numerade Educator
01:24

Problem 69

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.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:28

Problem 70

On average, 40 percent of U.S. beer drinkers order light beer. (a) What is the probability that none of the next eight customers who order beer will order light beer? (b) That one customer will? (c) Two customers? (d) Fewer than three? (e) Construct the probability distribution (Excel or Appendix A), make a graph of its PDF, and describe its shape.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:42

Problem 71

Write the Excel binomial formula for each probability.
a. Three successes in 20 trials with a 30 percent chance of success.
b. Seven successes in 50 trials with a 10 percent chance of success.
c. Six or fewer successes in 80 trials with a 5 percent chance of success.
d. At least 30 successes in 120 trials with a 20 percent chance of success.

Nick Johnson
Nick Johnson
Numerade Educator
03:08

Problem 72

Tired of careless spelling and grammar, a company decides to administer a test to all job applicants. The test consists of 20 sentences. Applicants must state whether each sentence contains any grammar or spelling errors. Half the sentences contain errors. The company requires a score of 14 or more. (a) If an applicant guesses randomly, what is the probability of passing? (b) What minimum score would be required to reduce the probability of "passing by guessing" to 5 percent or less?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:09

Problem 73

The default rate on government-guaranteed student loans at a certain private 4-year institution is 7 percent. The college extends 10 such loans. (a) What is the probability that none of them will default? (b) That at least three will default? (c) What is the expected number of defaults?

WM
William Mead
Numerade Educator
01:25

Problem 74

Experience indicates that 8 percent of the pairs of men's trousers dropped off for dry cleaning will have an object in the pocket that should be removed before cleaning. Suppose that 14 pairs of pants are dropped off and the cleaner forgets to check the pockets. What is the probability that no pair has an object in the pocket?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:59

Problem 75

A study by the Parents' Television Council showed that 80 percent of movie commercials aired on network television between 8 and 9 p.m. (the prime family viewing hour) were for R-rated films. (a) Find the probability that in 16 commercials during this time slot at least 10 will be for R-rated films. (b) Find the probability of fewer than 8 R-rated films.

Mahendra Kumar
Mahendra Kumar
Numerade Educator
00:52

Problem 76

Write the Excel formula for each Poisson probability, using a mean arrival rate of 10 arrivals per hour.
a. Seven arrivals. b. Three arrivals. c. Fewer than five arrivals. d. At least 11 arrivals.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:16

Problem 77

A small feeder airline knows that the probability is .10 that a reservation holder will not show up for its daily 7:15 a.m. flight into a hub airport. The flight carries 10 passengers. (a) If the flight is fully booked, what is the probability that all those with reservations will show up? (b) If the airline overbooks by selling 11 seats, what is the probability that no one will have to be bumped? (c) That more than one passenger will be bumped? *(d) The airline wants to overbook the flight by enough seats to ensure a 95 percent chance that the flight will be full, even if some passengers may be bumped. How many seats would it sell?

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
01:47

Problem 78

Although television HDTV converters are tested before they are placed in the installer's truck, the installer knows that 20 percent of them still won't work properly. The driver must install eight converters today in an apartment building. (a) Ten converters are placed in the truck. What is the probability that the driver will have enough working converters? *(b) How many boxes should the driver load to ensure a 95 percent probability of having enough working converters?

Sherrie Fenner
Sherrie Fenner
Numerade Educator
03:32

Problem 79

(a) Why might the number of calls received per minute at a fire station not be a Poisson event? (b) Name two other events per unit of time that might violate the assumptions of the Poisson model.

Mahnoor Khan
Mahnoor Khan
Numerade Educator
07:06

Problem 80

Software filters rely heavily on "blacklists" (lists of known "phishing" URLs) to detect fraudulent e-mails. But such filters typically catch only 20 percent of phishing URLs. Jason receives 16 phishing e-mails. (a) What is the expected number that would be caught by such a filter? (b) What is the chance that such a filter would detect none of them?

Jon Southam
Jon Southam
Numerade Educator

Problem 81

Lunch customers arrive at a Noodles \& Company restaurant at an average rate of 2.8 per minute. Define $X$ to be the number of customers to arrive during a randomly selected minute during the lunch hour and assume $X$ has a Poisson distribution. (a) Calculate the probability that exactly five customers will arrive in a minute during the lunch hour. (b) Calculate the probability that no more than five customers will arrive in a minute. (c) What is the average customer arrival rate for a 5-minute interval? (d) What property of the Poisson distribution did you use to find this arrival rate?

Check back soon!
01:49

Problem 82

In a major league baseball game, the average is 1.0 broken bat per game. Find the probability of (a) no broken bats in a game; (b) at least 2 broken bats.

Manisha Sarker
Manisha Sarker
Numerade Educator
01:59

Problem 83

In the last 50 years, the average number of deaths due to alligators in Florida is 0.3 death per year. Assuming no change in this average, in a given year find the probability of (a) no alligator deaths; (b) at least 2 alligator deaths.

AG
Ankit Gupta
Numerade Educator
02:38

Problem 84

In a recent year, potentially dangerous commercial aircraft incidents (e.g., near collisions) averaged 1.2 per 100,000 flying hours. Let $X$ be the number of incidents in a 100,000-hour period. (a) Justify the use of the Poisson model. (b) What is the probability of at least one incident? (c) More than three incidents? (d) Construct the probability distribution (Excel or Appendix B) and make a graph of its PDF.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:47

Problem 85

At an outpatient mental health clinic, appointment cancellations occur at a mean rate of 1.5 per day on a typical Wednesday. Let $X$ be the number of cancellations on a particular Wednesday. (a) Justify the use of the Poisson model. (b) What is the probability that no cancellations will occur on a particular Wednesday? (c) That one will? (d) More than two? (e) Five or more?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:49

Problem 86

Car security alarms go off at a mean rate of 3.8 per hour in a large Costco parking lot. Find the probability that in an hour there will be (a) no alarms; (b) fewer than four alarms; and (c) more than five alarms.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
10:39

Problem 87

In a certain automobile manufacturing paint shop, paint defects on the hood occur at a mean rate of 0.8 defect per square meter. A hood on a certain car has an area of 3 square meters. (a) Justify the use of the Poisson model. (b) If a customer inspects a hood at random, what is the probability that there will be no defects? (c) One defect? (d) Fewer than two defects?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:43

Problem 88

Past insurance company audits have found that 2 percent of dependents claimed on an employee's health insurance actually are ineligible for health benefits. An auditor examines a random sample of 7 claimed dependents. (a) What is the probability that all are eligible? (b) That at least one is ineligible?

Gus Steppen
Gus Steppen
Numerade Educator
04:38

Problem 89

A "rogue wave" (one far larger than others surrounding a ship) can be a threat to ocean-going vessels (e.g., naval vessels, container ships, oil tankers). The European Centre for Medium-Range Weather Forecasts issues a warning when such waves are likely. The average for this rare event is estimated to be .0377 rogue wave per hour in the South Atlantic. Find the probability that a ship will encounter at least one rogue wave in a 5-day South Atlantic voyage (120 hours).

Sophie Knight
Sophie Knight
Numerade Educator
01:09

Problem 90

In Northern Yellowstone Lake, earthquakes occur at a mean rate of 1.2 quakes per year. Let $X$ be the number of quakes in a given year. (a) Justify the use of the Poisson model. (b) What is the probability of fewer than three quakes? (c) More than five quakes? (d) Construct the probability distribution (Excel or Appendix B) and make a graph of its PDF

Akhil Choudhary
Akhil Choudhary
Numerade Educator
02:00

Problem 91

On New York's Verrazano Narrows bridge, traffic accidents occur at a mean rate of 2.0 crashes per day. Let $X$ be the number of crashes in a given day. (a) Justify the use of the Poisson model. (b) What is the probability of at least one crash? (c) Fewer than five crashes? (d) Construct the probability distribution (Excel or Appendix B), make a graph of its PDF, and describe its shape.

Kari Hasz
Kari Hasz
Numerade Educator
02:02

Problem 92

Leaks occur in a pipeline at a mean rate of 1 leak per 1,000 meters. In a 2,500-meter section of pipe, what is the probability of (a) no leaks? (b) Three or more leaks? (c) What is the expected number of leaks?

Narayan Hari
Narayan Hari
Numerade Educator
01:30

Problem 93

Among live deliveries, the probability of a twin birth is .02. (a) In 200 live deliveries, how many would be expected to have twin births? (b) What is the probability of no twin births? (c) One twin birth? (d) Calculate these probabilities both with and without an approximation. (e) Is the approximation justified? Discuss fully.

Foster Wisusik
Foster Wisusik
Numerade Educator
17:22

Problem 94

The probability is .03 that a passenger on United Airlines Flight 9841 is a Platinum flyer (50,000 miles per year). If 200 passengers take this flight, use Excel to find the binomial probability of (a) no Platinum flyers, (b) one Platinum flyer, and (c) two Platinum flyers. (d) Calculate the same probabilities using a Poisson approximation. (e) Is the Poisson approximation justified? Explain.

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
01:39

Problem 95

The probability of being "bumped" (voluntarily or involuntarily) on a U.S. airline was .00128. The average number of passengers traveling through Denver International Airport each hour is 5,708. (a) What is the expected number of bumped passengers per hour? (b) What is the approximate Poisson probability of fewer than 10 bumped passengers? More than 5 ? (c) Would you expect the approximation likely to be accurate (cite a rule of thumb)?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:28

Problem 96

On average, 2 percent of all persons who are given a breathalyzer test by the state police pass the test (blood alcohol under .08 percent). Suppose that 500 breathalyzer tests are given. (a) What is the expected number who pass the test? (b) What is the approximate Poisson probability that 5 or fewer will pass the test?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:43

Problem 97

The probability of an incorrect call by an NFL referee is .025 (e.g., calling a pass complete, but the decision reversed on instant replays). In a certain game, there are 150 plays. (a) What is the probability of at least 4 incorrect calls by the referees? (b) Justify any assumptions that you made.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
02:44

Problem 98

In CABG surgery, there is a .00014 probability of a retained foreign body (e.g., a sponge or a surgical instrument) left inside the patient. (a) In 100,000 CABG surgeries, what is the expected number of retained foreign bodies? (b) What is the Poisson approximation to the binomial probability of five or fewer retained foreign bodies in 100,000 CABG surgeries? (c) Look up CABG on the Internet if you are unfamiliar with the acronym. (See AHRQ News, No. 335, July 2008, p. 3.)

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:04

Problem 99

The probability of a job offer in a given interview is .25 . (a) What is the expected number of interviews until the first job offer? (b) What is the probability the first job offer occurs within the first six interviews?

Robin Corrigan
Robin Corrigan
Numerade Educator
View

Problem 100

The probability that a bakery customer will order a birthday cake is .04. (a) What is the expected number of customers until the first birthday cake is ordered? (b) What is the probability the first cake order occurs within the first 20 customers?

Keondre Parker
Keondre Parker
Numerade Educator
09:39

Problem 101

In a certain city, 8 percent of the cars have a burned-out headlight. (a) What is the expected number that must be inspected before the first one with a burned-out headlight is found? (b) What is the probability of finding the first one within the first five cars? Hint: Use the CDF.

Aparna Shakti
Aparna Shakti
Numerade Educator
01:46

Problem 102

For patients aged 81 to 90 , the probability is .07 that a coronary bypass patient will die soon after the surgery. (a) What is the expected number of operations until the first fatality? (b) What is the probability of conducting 20 or more operations before the first fatality? Hint: Use the CDF.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:10

Problem 103

Historically, 5 percent of a mail-order firm's regular charge-account customers have an incorrect current address in the firm's computer database. (a) What is the expected number of customer orders until the first one with an incorrect current address places an order? (b) What is the probability of mailing 30 bills or more until the first one is returned with a wrong address? Hint: Use the CDF.

Charles Machakwa
Charles Machakwa
Numerade Educator
03:05

Problem 104

At a certain clinic, 2 percent of all pap smears show signs of abnormality. What is the expected number of pap smears that must be inspected before the first abnormal one is found?

Sherrie Fenner
Sherrie Fenner
Numerade Educator
00:37

Problem 105

The weight of a Los Angeles Lakers basketball player averages 233.1 pounds with a standard deviation of 34.95 pounds. To express these measurements in terms a European would understand, we could convert from pounds to kilograms by multiplying by .4536 . (a) In kilograms, what is the mean? (b) In kilograms, what is the standard deviation?

Nick Johnson
Nick Johnson
Numerade Educator
02:43

Problem 106

The Rejuvo Corp. manufactures granite countertop cleaner and polish. Quarterly sales $Q$ is a random variable with a mean of 25,000 bottles and a standard deviation of 2,000 bottles. Variable cost is $$\$ 8$$ per unit and fixed cost is $$\$ 150,000$$. (a) Find the mean and standard deviation of Rejuvo's total cost. (b) If all bottles are sold, what would the selling price have to be to break even, on average? To make a profit of $$\$ 20,000$$ ?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:05

Problem 107

A manufacturer fills one-gallon cans $(3,785 \mathrm{ml})$ on an assembly line in two independent steps. First, a high-volume spigot injects most of the paint rapidly. Next, a more precise but slower spigot tops off the can. The fill amount in each step is a normally distributed random variable. For step one, $\mu_1=3,420 \mathrm{ml}$ and $\sigma_2=10 \mathrm{ml}$, while for step two $\mu_2=390 \mathrm{ml}$ and $\sigma_2=2 \mathrm{ml}$. Find the mean and standard deviation of the total fill $X_1+X_2$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:29

Problem 108

A manufacturing project has five independent phases whose completion must be sequential. The time to complete each phase is a random variable. The mean and standard deviation of the time for each phase are shown below. (a) Find the expected completion time. (b) Make a 2 -sigma interval around the mean completion time $(\mu \pm 2 \sigma)$.
$$
\begin{array}{lcc}
\hline \text { Phase } & \text { Mean (hours) } & \text { Std. Dev. (hours) } \\
\hline \text { Set up dies and other tools } & 20 & 4 \\
\text { Milling and machining } & 10 & 2 \\
\text { Finishing and painting } & 14 & 3 \\
\text { Packing and crating } & 6 & 2 \\
\text { Shipping } & 48 & 6 \\
\hline
\end{array}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:01

Problem 109

109 In September, demand for industrial furnace boilers at a large plumbing supply warehouse has a mean of 7 boilers with a standard deviation of 2 boilers. The warehouse pays a unit cost of $$\$ 2,225$$ per boiler plus a fee of $$\$ 500$$ per month to act as dealer for these boilers. Boilers are sold for $$\$ 2,850$$ each. (a) Find the mean and standard deviation of September profit (revenue minus cost).
(b) Which rules did you use?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:27

Problem 110

A certain outpatient medical procedure has five steps that must be performed in sequence. (a) Assuming that the time (in minutes) required for each step is an independent random variable, find the mean and standard deviation for the total time. (b) Why might the assumption of independence be doubtful?
$$
\begin{array}{lcc}
\hline \text { Step } & \text { Mean (minutes) } & \text { Standard Deviation (minutes) } \\
\hline \text { Patient check-in } & 15 & 4 \\
\text { Pre-op preparation } & 30 & 6 \\
\text { Medical procedure } & 25 & 5 \\
\text { Recovery } & 45 & 10 \\
\text { Check-out and discharge } & 20 & 5 \\
\hline
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 111

Malaprop Ltd. sells two products. Daily sales of product $A$ have a mean of $$\$ 70$$ with a standard deviation of $$\$ 10$$, while sales of product $B$ have a mean of $$\$ 200$$ with a standard deviation of $$\$ 30$$. Sales of the products tend to rise and fall at the same time, having a positive covariance of 400 . (a) Find the mean daily sales for both products together. (b) Find the standard deviation of total sales of both products. (c) Is the variance of the total sales greater than or less than the sum of the variances for the two products?

Sheryl Ezze
Sheryl Ezze
Numerade Educator