Statistics Informed Decisions Using Data

Michael Sullivan III

Chapter 5

Probability - all with Video Answers

Educators

+ 2 more educators

Section 1

Probability Rules

Problem 1

What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean the event is impossible?

Sherrie Fenner

Sherrie Fenner

Numerade Educator

Problem 2

What does it mean for an event to be unusual? Why should the cutoff for identifying unusual events not always be $0.05 ?$

Sherrie Fenner

Numerade Educator

Problem 3

True or False: In a probability model, the sum of the probabilities of all outcomes must equal 1 .

Sherrie Fenner

Numerade Educator

Problem 4

True or False: Probability is a measure of the likelihood of a random phenomenon or chance behavior.

Sherrie Fenner

Numerade Educator

Problem 5

In probability, $a(n)$ _______ is any process that can be repeated in which the results are uncertain.

Manisha Sarker

Numerade Educator

Problem 6

$\mathrm{A}(\mathrm{n}) $ _____ is any collection of outcomes from a probability experiment.

Sherrie Fenner

Numerade Educator

Problem 7

Verify that the table in the next column is a probability model. What do we call the outcome "blue"?

Rashmi Sinha

Numerade Educator

Problem 8

Verify that the following is a probability model. If the model represents the colors of M\&Ms in a bag of milk chocolate M\&Ms, explain what the model implies.

Nick Johnson

Numerade Educator

Problem 9

Why is the following not a probability model?
$$
\begin{array}{lc}
\text { Color } & \text { Probability } \\
\hline \text { Red } & 0.3 \\
\hline \text { Green } & -0.3 \\
\hline \text { Blue } & 0.2 \\
\hline \text { Brown } & 0.4 \\
\hline \text { Yellow } & 0.2 \\
\hline \text { Orange } & 0.2 \\
\hline
\end{array}
$$

Emily Himsel

Numerade Educator

Problem 10

Why is the following not a probability model?
$$
\begin{array}{lc}
\text { Color } & \text { Probability } \\
\hline \text { Red } & 0.1 \\
\hline \text { Green } & 0.1 \\
\hline \text { Blue } & 0.1 \\
\hline \text { Brown } & 0.4 \\
\hline \text { Yellow } & 0.2 \\
\hline \text { Orange } & 0.3
\end{array}
$$

Emily Himsel

Numerade Educator

Problem 11

Which of the following numbers could be the probability of an event?
$$
0,0.01,0.35,-0.4,1,1.4
$$

Sherrie Fenner

Numerade Educator

Problem 12

Which of the following numbers could be the probability of an event?
$$
1.5, \frac{1}{2}, \frac{3}{4}, \frac{2}{3}, 0,-\frac{1}{4}
$$

Sherrie Fenner

Numerade Educator

Problem 13

According to Nate Silver, the probability of a senate candidate winning his/her election with a $5 \%$ lead in an average of polls with a week until the election is $0.89 .$ Interpret this probability.

Rashmi Sinha

Numerade Educator

Problem 14

In seven-card stud poker, a player is dealt seven cards. The probability that the player is dealt two cards of the same value and five other cards of different value so that the player has a pair is $0.44 .$ Explain what this probability means. If you play seven-card stud 100 times, will you be dealt a pair exactly 44 times? Why or why not?

Sherrie Fenner

Numerade Educator

Problem 15

Suppose that you toss a coin 100 times and get 95 heads and five tails. Based on these results, what is the estimated probability that the next flip results in a head?

Anna D.

Numerade Educator

Problem 16

Suppose that you roll a die 100 times and get six 80 times. Based on these results, what is the estimated probability that the next roll results in six?

Sherrie Fenner

Numerade Educator

Problem 17

Bob is asked to construct a probability model for rolling a pair of fair dice. He lists the outcomes as $2,3,4,5,6,7,8,$ $9,10,11,12 .$ Because there are 11 outcomes, he reasoned, the probability of rolling a two must be $\frac{1}{11} .$ What is wrong with Bob's reasoning?

Manisha Sarker

Numerade Educator

Problem 18

A person can have one of four blood types: A, B, AB, or O. If a person is randomly selected, is the probability they have blood type A equal to $\frac{1}{4} ?$ Why?

Jacob Fry

Numerade Educator

Problem 19

If a person rolls a six-sided die and then flips a coin, describe the sample space of possible outcomes using 1,2,3,4,5,6 for the die outcomes and $\mathrm{H}, \mathrm{T}$ for the coin outcomes.

Sherrie Fenner

Numerade Educator

Problem 20

If a basketball player shoots three free throws, describe the sample space of possible outcomes using $S$ for a made free throw and $F$ for a missed free throw.

Sherrie Fenner

Numerade Educator

Problem 21

According to the U.S. Department of Education, $42.8 \%$ of 3-year-olds are enrolled in day care. What is the probability that a randomly selected 3-year-old is enrolled in day care?

Sherrie Fenner

Numerade Educator

Problem 22

According to the American Veterinary Medical Association, the proportion of households owning a dog is $0.372 .$ What is the probability that a randomly selected household owns a dog?

Sherrie Fenner

Numerade Educator

Problem 23

Let the sample space be $S=\{1,2,3,4,5,6,$ 7,8,9,10\}$.$ Suppose the outcomes are equally likely.
Compute the probability of the event $E=\{1,2,3\}$.

Trinity Steen

Numerade Educator

Problem 24

Let the sample space be $S=\{1,2,3,4,5,6,$ 7,8,9,10\}$.$ Suppose the outcomes are equally likely.
Compute the probability of the event $F=\{3,5,9,10\}$.

Trinity Steen

Numerade Educator

Problem 25

Let the sample space be $S=\{1,2,3,4,5,6,$ 7,8,9,10\}$.$ Suppose the outcomes are equally likely.
Compute the probability of the event $E="$ an even number less than 9 "

Nick Johnson

Numerade Educator

Problem 25

Let the sample space be $S=\{1,2,3,4,5,6,$ 7,8,9,10\}$.$ Suppose the outcomes are equally likely.
Compute the probability of the event $E="$ an even number less than 9."

Nick Johnson

Numerade Educator

Problem 26

Let the sample space be $S=\{1,2,3,4,5,6,$ 7,8,9,10\}$.$ Suppose the outcomes are equally likely.
Compute the probability of the event $F="$ an odd number."

Trinity Steen

Numerade Educator

Problem 27

A survey of 500 randomly selected high school students determined that 288 played organized sports.
(a) What is the probability that a randomly selected high school student plays organized sports?
(b) Interpret this probability.

Jacob Fry

Numerade Educator

Problem 28

In a survey of 1100 female adults (18 years of age or older), it was determined that 341 volunteered at least once in the past year.
(a) What is the probability that a randomly selected adult female volunteered at least once in the past year?
(b) Interpret this probability.

Jacob Fry

Numerade Educator

Problem 29

The Wall Street Journal regularly publishes an article entitled "The Count." In one article, The Count looked at 1000 randomly selected home runs in Major League Baseball.
(a) Of the 1000 homeruns, it was found that 85 were caught by fans. What is the probability that a randomly selected homerun is caught by a fan?
(b) Of the 1000 homeruns, it was found that 296 were dropped when a fan had a legitimate play on the ball. What is the probability that a randomly selected homerun is dropped?
(c) Of the 85 caught balls, it was determined that 34 were barehanded catches, 49 were caught with a glove, and two were caught in a hat. What is the probability a randomly selected caught ball was caught in a hat? Interpret this probability.
(d) Of the 296 dropped balls, it was determined that 234 were barehanded attempts, 54 were dropped with a glove, and eight were dropped with a failed hat attempt. What is the probability a randomly selected dropped ball was a failed hat attempt? Interpret this probability.

Nick Johnson

Numerade Educator

Problem 30

A bag of 100 tulip bulbs purchased from a nursery contains 40 red tulip bulbs, 35 yellow tulip bulbs, and 25 purple tulip bulbs.
(a) What is the probability that a randomly selected tulip bulb is red?
(b) What is the probability that a randomly selected tulip bulb is purple?
(c) Interpret these two probabilities.

Jacob Fry

Numerade Educator

Problem 31

In the game of roulette, a wheel consists of 38 slots numbered $0,00,1,2, \ldots, 36$. (See the photo.) To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots.
(a) Determine the sample space.
(b) Determine the probability that the metal ball falls into the slot marked eight. Interpret this probability.
(c) Determine the probability that the metal ball lands in an odd slot. Interpret this probability.

Kari Hasz

Numerade Educator

Problem 32

Birthdays Exclude leap years from the following calculations and assume each birthday is equally likely:
(a) Determine the probability that a randomly selected person has a birthday on the 1 st day of a month. Interpret this probability.
(b) Determine the probability that a randomly selected person has a birthday on the 31 st day of a month. Interpret this probability.
(c) Determine the probability that a randomly selected person was born in December. Interpret this probability.
(d) Determine the probability that a randomly selected person has a birthday on November 8 . Interpret this probability.
(e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct?
(f) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December?

Gretchen Laviolette

Gretchen Laviolette

Numerade Educator

Problem 33

Genetics A gene is composed of two alleles, either dominant or recessive. Suppose that a husband and wife, who are both carriers of the sickle-cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal-cell allele $(S)$ and one recessive sickle-cell allele $(s) .$ Therefore, the genotype of each parent is Ss. Each parent contributes one allele to his or her offspring, with each allele being equally likely.
(a) List the possible genotypes of their offspring.
(b) What is the probability that the offspring will have sicklecell anemia? In other words, what is the probability that the offspring will have genotype $s s ?$ Interpret this probability.
(c) What is the probability that the offspring will not have sickle-cell anemia but will be a carrier? In other words, what is the probability that the offspring will have one dominant normal-cell allele and one recessive sickle-cell allele? Interpret this probability.

Kari Hasz

Numerade Educator

Problem 34

More Genetics In Problem $33,$ we learned that for some diseases, such as sickle-cell anemia, an individual will get the disease only if he or she receives both recessive alleles. This is not always the case. For example, Huntington's disease only requires one dominant gene for an individual to contract the disease. Suppose that a husband and wife, who both have a dominant Huntington's disease allele $(S)$ and a normal recessive allele $(s),$ decide to have a child.
(a) List the possible genotypes of their offspring.
(b) What is the probability that the offspring will not have Huntington's disease? In other words, what is the probability that the offspring will have genotype ss? Interpret this probability.
(c) What is the probability that the offspring will have Huntington's disease?

Kari Hasz

Numerade Educator

Problem 35

In a national survey conducted by the Centers for Disease Control to determine college students' health-risk behaviors, college students were asked, "How often do you wear a seatbelt when riding in a car driven by someone else?" The frequencies appear in the following table:
(a) Construct a probability model for seatbelt use by a passenger.
(b) Would you consider it unusual to find a college student who never wears a seatbelt when riding in a car driven by someone else? Why?

Kari Hasz

Numerade Educator

Problem 36

In a national survey conducted by the Centers for Disease Control to determine college students' health-risk behaviors, college students were asked, "How often do you wear a seatbelt when driving a car?" The frequencies appear in the following table:
$$
\begin{array}{lc}
\text { Response } & \text { Frequency } \\
\hline \text { Never } & 118 \\
\hline \text { Rarely } & 249 \\
\hline \text { Sometimes } & 345 \\
\hline \text { Most of the time } & 716 \\
\hline \text { Always } & 3093
\end{array}
$$
(a) Construct a probability model for seatbelt use by a driver.
(b) Is it unusual for a college student to never wear a seatbelt when driving a car? Why?

Jacob Fry

Numerade Educator

Problem 37

A police officer randomly selected 642 police records of larceny thefts. The following data represent the number of offenses for various types of larceny thefts.
$$
\begin{array}{lc}
\text { Type of Larceny Theft } & \text { Number of Offenses } \\
\hline \text { Pocket picking } & 4 \\
\hline \text { Purse snatching } & 6 \\
\hline \text { Shoplifting } & 133 \\
\hline \text { From motor vehicles } & 219 \\
\hline \text { Motor vehicle accessories } & 90 \\
\hline \text { Bicycles } & 42 \\
\hline \text { From buildings } & 143 \\
\hline \text { From coin-operated } & 5 \\
\text { machines } &
\end{array}
$$
(a) Construct a probability model for type of larceny theft.
(b) Are purse snatching larcenies unusual?
(c) Are bicycle larcenies unusual?

Kari Hasz

Numerade Educator

Problem 38

The following data represent the number of live multiple-delivery births (three or more babies) in 2012 for women 15 to 54 years old.
$$
\begin{array}{lc}
\text { Age } & \text { Number of Multiple Births } \\
\hline 15-19 & 44 \\
\hline 20-24 & 404 \\
\hline 25-29 & 1204 \\
\hline 30-34 & 1872 \\
\hline 35-39 & 1000 \\
\hline 40-44 & 332 \\
\hline 45-54 & 63
\end{array}
$$
(a) Construct a probability model for number of multiple births.
(b) In the sample space of all multiple births, are multiple births for 15 - to 19 -year-old mothers unusual?
(c) In the sample space of all multiple births, are multiple births for 40 - to 44 -year-old mothers unusual?

Donna Densmore

Numerade Educator

Problem 39

Use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to answer the following questions.
Which of the assignments of probabilities are consistent with the definition of a probability model?

Manisha Sarker

Numerade Educator

Problem 40

Use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to answer the following questions.
Which of the assignments of probabilities should be used if the coin is known to be fair?

Manisha Sarker

Numerade Educator

Problem 41

Use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to answer the following questions.
Which of the assignments of probabilities should be used if the coin is known to always come up tails?

Manisha Sarker

Numerade Educator

Problem 42

Use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to answer the following questions.
Which of the assignments of probabilities should be used if tails is twice as likely to occur as heads?

Manisha Sarker

Numerade Educator

Problem 43

Going to Disney World John, Roberto, Clarice, Dominique, and Marco work for a publishing company. The company wants to send two employees to a statistics conference in Orlando. To be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat.
(a) Determine the sample space of the experiment. That is, list all possible simple random samples of size $n=2$.
(b) What is the probability that Clarice and Dominique attend the conference?
(c) What is the probability that Clarice attends the conference?
(d) What is the probability that John stays home?

Sherrie Fenner

Numerade Educator

Problem 44

In 2011, Six Flags St. Louis had 10 roller coasters:
The Screamin' Eagle, The Boss, River King Mine Train, Batman the Ride, Mr. Freeze, Ninja, Tony Hawk's Big Spin, Evel Knievel, Xcalibur, and Sky Screamer. Of these, The Boss, The Screamin' Eagle, and Evel Knievel are wooden coasters. Ethan wants to ride two more roller coasters before leaving the park (not the same one twice) and decides to select them by drawing names from a hat.
(a) Determine the sample space of the experiment. That is, list all possible simple random samples of size $n=2$.
(b) What is the probability that Ethan will ride $\mathrm{Mr}$. Freeze and Evel Knievel?
(c) What is the probability that Ethan will ride the Screamin' Eagle?
(d) What is the probability that Ethan will ride two wooden roller coasters?
(e) What is the probability that Ethan will not ride any wooden roller coasters?

Kari Hasz

Numerade Educator

Problem 45

On October 5,2001 , Barry Bonds broke Mark McGwire's homerun record for a single season by hitting his 71 st and 72 nd homeruns. Bonds went on to hit one more homerun before the season ended, for a total of $73 .$ Of the 73 homeruns, 24 went to right field, 26 went to right center field, 11 went to center field, 10 went to left center field, and 2 went to left field. Source: Baseball-almanac.com
(a) What is the probability that a randomly selected homerun was hit to right field?
(b) What is the probability that a randomly selected homerun was hit to left field?
(c) Was it unusual for Barry Bonds to hit a homerun to left field? Explain.

Kari Hasz

Numerade Educator

Problem 46

Rolling a Die
(a) Roll a single die 50 times, recording the result of each roll of the die. Use the results to approximate the probability of rolling a three.
(b) Roll a single die 100 times, recording the result of each roll of the die. Use the results to approximate the probability of rolling a three.
(c) Compare the results of (a) and (b) to the classical probability of rolling a three.

Sherrie Fenner

Numerade Educator

Problem 47

Use a graphing calculator or statistical software to simulate rolling a six-sided die 100 times, using an integer distribution with numbers one through six.
(a) Use the results of the simulation to compute the probability of rolling a one.
(b) Repeat the simulation. Compute the probability of rolling a one.
(c) Simulate rolling a six-sided die 500 times. Compute the probability of rolling a one.
(d) Which simulation resulted in the closest estimate to the probability that would be obtained using the classical method?

Sherrie Fenner

Numerade Educator

Problem 48

Determine whether the probabilities below are computed using classical methods, empirical methods, or subjective methods.
(a) The probability of having eight girls in an eight-child family is $0.00390625 .$
(b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is $0.0054 .$
(c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about 0.30 .
(d) On the basis of clinical trials, the probability of efficacy of a new drug is 0.75 .

Kari Hasz

Numerade Educator

Problem 49

You suspect a 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the table below. Do you think the die is loaded? Why?
$$
\begin{array}{cc}
\text { Value of Die } & \text { Frequency } \\
\hline 1 & 105 \\
\hline 2 & 47 \\
\hline 3 & 44 \\
\hline 4 & 49 \\
\hline 5 & 51 \\
\hline 6 & 104 \\
\hline
\end{array}
$$

Kari Hasz

Numerade Educator

Problem 50

Conduct a survey in your school by randomly asking 50 students whether they drive to school. Based on the results of the survey, approximate the probability that a randomly selected student drives to school.

Bryan Meares

Numerade Educator

Problem 51

In 2013 , the median income of families in the United States was $\$ 52,250 .$ What is the probability that a randomly selected family has an income greater than $\$ 52,250 ?$

Jacob Fry

Numerade Educator

Problem 52

The middle $50 \%$ of enrolled freshmen at Washington University in St. Louis had SAT math scores in the range $700-780 .$ What is the probability that a randomly selected freshman at Washington University has an SAT math score of 700 or higher?

Sherrie Fenner

Numerade Educator

Problem 53

Each year the National Football League (NFL) runs a combine in which players who wish to be considered for the NFL draft must participate in a variety of activities. Go to www.pearsonhighered.com/sullivanstats to obtain the data file $5_{-} 1_{-} 53$ using the file format of your choice for the version of the text you are using. The data represent results of the 2015 NFL combine. Construct a probability model for the variable "POS," which represents the position of the player. For example, Nelson Ahholor plays wide receiver (WR). If a player is randomly selected from the 2015 NFL combine, what position has the highest probability of being selected? Would you be surprised if a center (C) was randomly selected? Why?

Rashmi Sinha

Numerade Educator

Problem 54

Drug Side Effects In placebocontrolled clinical trials for the drug Viagra, 734 subjects received Viagra and 725 subjects received a placebo (subjects did not know which treatment they received). The table in the next column summarizes reports of various side effects that were reported.

Kari Hasz

Numerade Educator

Problem 55

Explain the Law of Large Numbers. How does this law apply to gambling casinos?

Sherrie Fenner

Numerade Educator

Problem 56

In computing classical probabilities, all outcomes must be equally likely. Explain what this means.

Sherrie Fenner

Numerade Educator

Problem 57

Describe what an unusual event is. Should the same cutoff always be used to identify unusual events? Why or why not?

Sherrie Fenner

Numerade Educator

Problem 58

You are planning a trip to a water park tomorrow and the weather forecaster says there is a $70 \%$ chance of rain. Explain what this result means.

Sherrie Fenner

Numerade Educator

Problem 59

Describe the difference between classical and empirical probability.

Sherrie Fenner

Numerade Educator

Problem 60

In a September 19,2010 , article in Parade Magazine written to Ask Marilyn, Marilyn vos Savant was asked the following: Four identical sealed envelopes are on a table, one of which contains $\$ 100 .$ You are to select one of the envelopes. Then the game host discards two of the remaining three envelopes and informs you that they do not contain the $\$ 100 .$ In addition, the host offers you the opportunity to switch envelopes. What should you do?
(a) Keep your envelope
(b) switch
(c) it does not matter.

Sherrie Fenner

Numerade Educator

Problem 61

Suppose you live in a town with two hospitals - one large and the other small. On a given day in one of the hospitals, $60 \%$ of the babies who were born were girls. Which one do you think it is? Or, is it impossible to tell. Support your decision?

Kari Hasz

Numerade Educator