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STATS Modeling The World

David E. Bock, Paul F. Velleman, Richard D. De Veaux

Chapter 4

Randomness and Probability

Educators

SC

Problem 1

For each of the following, list the sample space and tell whether you think the events are equally likely:
a) Toss 2 coins; record the order of heads and tails.
b) A family has 3 children; record the number of boys.
c) Flip a coin until you get a head or 3 consecutive tails; record each flip.
d) Roll two dice; record the larger number.

SC
Stephanie C.
Numerade Educator

Problem 2

For each of the following, list the sample space and tell whether you think the events are equally likely:
a) Roll two dice; record the sum of the numbers.
b) A family has 3 children; record each child’s sex in order of birth.
c) Toss four coins; record the number of tails.
d) Toss a coin 10 times; record the length of the longest run of heads.

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Problem 3

A casino claims that its roulette wheel is truly random. What should that claim mean?

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Problem 4

The weather reporter on TV makes predictions such as a 25% chance of rain. What do you think is the meaning of such a phrase?

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Problem 5

Comment on the following quotation:
“What I think is our best determination is it will be a colder than normal winter,” said Pamela Naber Knox, a Wisconsin state climatologist. “I’m basing that on a couple of different things. First, in looking at the past few winters, there has been a lack of really cold weather. Even though we are not supposed to use the law of averages, we are due.” (Associated Press, fall 1992, quoted by Schaeffer et al.)

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Problem 6

After an unusually dry autumn, a radio announcer is heard to say, “Watch out! We’ll pay for these sunny days later on this winter.” Explain what he’s trying to say, and comment on the validity of his reasoning.

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Problem 7

A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward, he says he was very confident that last time at bat because he knew he was “due for a hit.” Comment on his reasoning.

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Problem 8

Commercial airplanes have an excellent safety record. Nevertheless, there are crashes occasionally, with the loss of many lives. In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying.
a) A travel agent suggests that since the law of averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think?
b) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash?

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Problem 9

Insurance companies collect annual payments from drivers in exchange for paying for the cost of accidents.
a) Why should you be reluctant to accept a $\$ 1500$ payment from your neighbor to cover his automobile accidents in the next year?
b) Why can the insurance company make that offer?

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Problem 10

On February 11, 2009, the AP news wire released the following story:
(LAS VEGAS, Nev.)—A man in town to watch the NCAA basketball tournament hit a $\$ 38.7$ million jackpot on Friday, the biggest slot machine payout ever. The 25 -year-old software engineer from Los Angeles, whose name was not released at his request, won after putting three $\$ 1$ coins in a machine at the Excalibur hotel-casino, said Rick Sorensen, a spokesman for slot machine maker International Game Technology.
a) How can the Excalibur afford to give away millions of dollars on a $\$ 3$ bet?
b) Why was the maker willing to make a statement? Wouldn't most businesses want to keep such a huge loss quiet?

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Problem 11

In your dresser are five blue shirts, three red shirts, and two black shirts.
a) What is the probability of randomly selecting a red shirt?
b) What is the probability that a randomly selected shirt is not black?

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Problem 12

Your list of favorite songs contains 10 rock songs, 7 rap songs, and 3 country songs.
a) What is the probability that a randomly played song is a rap song?
b) What is the probability that a randomly played song is not country?

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Problem 13

A 2010 study conducted by the National Center for Health Statistics found that 25% of U.S. households had no landline service. This raises concerns about the accuracy of certain surveys, as they depend on random-digit dialing to households via landlines. We are going to pick five U.S. households at random:
a) What is the probability that all five of them have a landline?
b) What is the probability that at least one of them does not have a landline?
c) What is the probability that at least one of them does have a landline?

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Problem 14

The survey by the National Center for Health Statistics further found that 49% of adults ages 25–29 had only a cell phone and no landline. We randomly select four 25–29-year-olds:
a) What is the probability that all of these adults have a only a cell phone and no landline?
b) What is the probability that none of these adults have only a cell phone and no landline?
c) What is the probability that at least one of these adults has only a cell phone and no landline?

Tony W.
Numerade Educator

Problem 15

The plastic arrow on a spinner for a child’s game stops rotating to point at a color that will determine what happens next. Which of the following probability assignments are possible?

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Problem 16

Many stores run “secret sales”: Shoppers receive cards that determine how large a discount they get, but the percentage is revealed by scratching off that black stuff (what is that?) only after the purchase has been totaled at the cash register. The store is required to reveal (in the fine print) the distribution of discounts available. Which of these probability assignments are legitimate?

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Problem 17

Suppose that 46% of families living in a certain county own a computer and 18% own an HDTV. The Addition Rule might suggest, then, that 64% of families own either a computer or an HDTV. What’s wrong with that reasoning?

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Problem 18

Funding for many schools comes from taxes based on assessed values of local properties. People’s homes are assessed higher if they have extra features such as garages and swimming pools. Assessment records in a certain school district indicate that 37% of the homes have garages and 3% have swimming pools. The Addition Rule might suggest, then, that 40% of residences have a garage or a pool. What’s wrong with that reasoning?

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Problem 19

Traffic checks on a certain section of highway suggest that 60$\%$ of drivers are speeding there. Since $0.6 \times 0.6=0.36,$ the Multiplication Rule might suggest that there's a 36$\%$ chance that two vehicles in a row are both speeding. What's wrong with that reasoning?

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Problem 20

Although it's hard to be definitive in classifying people as right- or left-handed, some studies suggest that about 14$\%$ of people are left-handed. Since $0.14 \times 0.14=0.0196,$ the Multiplication Rule might suggest that there's about a 2$\%$ chance that a brother and a sister are both lefties. What's wrong with that reasoning?

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Problem 21

For high school students graduating in 2007, college admissions to the nation’s most selective schools were the most competitive in memory. (The New York Times, “A Great Year for Ivy League Schools, but Not So Good for Applicants to Them,” April 4, 2007). Harvard accepted about 9% of its applicants, Stanford 10%, and Penn 16%. Jorge has applied to all three. Assuming that he’s a typical applicant, he figures that his chances of getting into both Harvard and Stanford must be about 0.9%.
a) How has he arrived at this conclusion?
b) What additional assumption is he making?
c) Do you agree with his conclusion?

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Problem 22

In Exercise 21, we saw that in 2007 Harvard accepted about 9% of its applicants, Stanford 10%, and Penn 16%. Jorge has applied to all three. He figures that his chances of getting into at least one of the three must be about 35%.
a) How has he arrived at this conclusion?
b) What assumption is he making?
c) Do you agree with his conclusion?

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Problem 23

A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require three or more repairs. What is the probability that a car chosen at random will need
a) no repairs?
b) no more than one repair?
c) some repairs?

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Problem 24

Your family has an iPod filled with music. It has many thousands of songs. You figure that roughly 60% of the songs are music you like, 25% of the music is annoying songs from your little sister, and the rest is stuff from the 80s that only your parents still think is cool. Driving across town, your Mom puts the iPod on shuffle and you all listen to whatever randomness produces. What is the probability that the first song is
a) an 80s song?
b) a song picked by one of the kids?
c) not one of your songs?

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Problem 25

Consider again the auto repair rates described in Exercise 23. If you own two cars, what is the probability that
a) neither will need repair?
b) both will need repair?
c) at least one car will need repair?

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Problem 26

You listen to two songs, as described in Exercise 24. What is the probability that
a) neither will be songs that you like?
b) both will be annoying songs from your sister?
c) at least one will be a song of your Mom’s choice?

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Problem 27

You used the Multiplication Rule to calculate repair probabilities for your cars in Exercise 25.
a) What must be true about your cars in order to make that approach valid?
b) Do you think this assumption is reasonable? Explain.

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Problem 28

You used the Multiplication Rule to calculate probabilities about the music choices of your iPod in Exercise 26.
a) What must be true about the songs to make that approach valid?
b) Do you think this assumption is reasonable? Explain.

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Problem 29

A Gallup Poll in March 2011 asked 1012 U.S. adults whether increasing domestic energy production or protecting the environment should be given a higher priority. Here are the results:
If we select a person at random from this sample of 1012 adults,
a) what is the probability that the person responded “Increase production”?
b) what is the probability that the person responded “Equally important” or had no opinion?

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Problem 30

A Pew Research poll in 2011 asked 2005 U.S. adults whether being a father today is harder than it was a generation ago. Here’s how they responded:
If we select a respondent at random from this sample of 2005 adults,
a) what is the probability that the selected person responded “Harder”?
b) what is the probability that the person responded the “Same” or “Easier”?

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Problem 31

Exercise 29 shows the results of a Gallup Poll about energy. Suppose we select three people at random from this sample.
a) What is the probability that all three responded “Protect the environment”?
b) What is the probability that none responded “Equally important”?
c) What assumption did you make in computing these probabilities?
d) Explain why you think that assumption is reasonable.

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Problem 32

Consider again the results of the poll about fathering discussed in Exercise 30. If we select two people at random from this sample,
a) what is the probability that both think that being a father is easier today?
b) what is the probability that neither thinks being a father is easier today?
c) what is the probability that the first person thinks being a father is easier today and the second one doesn’t?
d) what assumption did you make in computing these probabilities?
e) explain why you think that assumption is reasonable.

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Problem 33

As mentioned in the chapter, opinion-polling organizations contact their respondents by sampling random telephone numbers. Although interviewers now can reach about 62% of U.S. households, the percentage of those contacted who agree to cooperate with the survey has fallen from 43% in 1997 to only 14% in 2012 (Pew Research Center for the People and the Press). Each household, of course, is independent of the others.
a) What is the probability that the next household on the list will be contacted but will refuse to cooperate?
b) What was the probability (in 2012) of failing to contact a household or of contacting the household but not getting them to agree to the interview?
c) Show another way to calculate the probability in part b.

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Problem 34

According to Pew Research, the contact rate (probability of contacting a selected household) was 90% in 1997 and 62% in 2012. However, the cooperation rate (probability of someone at the contacted household agreeing to be interviewed) was 43% in 1997 and dropped to 14% in 2012.
a) What was the probability (in 2012) of obtaining an interview with the next household on the sample list? (To obtain an interview, an interviewer must both contact the household and then get agreement for the interview.)
b) Was it more likely to obtain an interview from a randomly selected household in 1997 or in 2012?

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Problem 35

The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&M’s, red another 20%, and orange, blue, and green each made up 10%. The rest were brown.
a) If you pick an M&M at random, what was the prob-ability that
1) it is brown?
2) it is yellow or orange?
3) it is not green?
4) it is striped?
b) If you pick three M&M’s in a row, what is the prob- ability that
1) they are all brown?
2) the third one is the first one that’s red?
3) none are yellow?
4) at least one is green?

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Problem 36

The American Red Cross says that about 45% of the U.S. population has Type O blood, 40% Type A, 11% Type B, and the rest Type AB.
a) Someone volunteers to give blood. What is the probability that this donor
1) has Type AB blood?
2) has Type A or Type B?
3) is not Type O?
b) Among four potential donors, what is the probability that
1) all are Type O?
2) no one is Type AB?
3) they are not all Type A?
4) at least one person is Type B?

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Problem 37

In Exercise 35 you calculated probabilities of getting various M&M’s. Some of your answers depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesn’t affect the probability of the other. Do you understand the difference between disjoint and independent?
a) If you draw one M&M, are the events of getting a red one and getting an orange one disjoint, independent, or neither?
b) If you draw two M&M’s one after the other, are the events of getting a red on the first and a red on the second disjoint, independent, or neither?
c) Can disjoint events ever be independent? Explain.

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Problem 38

In Exercise 36 you calculated probabilities involving various blood types. Some of your answers depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesn’t affect the probability of the other. Do you understand the difference between disjoint and independent?
a) If you examine one person, are the events that the person is Type A and that the person is Type B disjoint, independent, or neither?
b) If you examine two people, are the events that the first is Type A and the second Type B disjoint, independent, or neither?
c) Can disjoint events ever be independent? Explain.

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Problem 39

You roll a fair die three times. What is the probability that
a) you roll all 6’s?
b) you roll all odd numbers?
c) none of your rolls gets a number divisible by 3?
d) you roll at least one 5?
e) the numbers you roll are not all 5’s?

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Problem 40

A slot machine has three wheels that spin independently. Each has 10 equally likely symbols: 4 bars, 3 lemons, 2 cherries, and a bell. If you play, what is the probability that
a) you get 3 lemons?
b) you get no fruit symbols?
c) you get 3 bells (the jackpot)?
d) you get no bells?
e) you get at least one bar (an automatic loser)?

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Problem 41

A certain bowler can bowl a strike 70% of the time. What’s the probability that she
a) goes three consecutive frames without a strike?
b) makes her first strike in the third frame?
c) has at least one strike in the first three frames?
d) bowls a perfect game (12 consecutive strikes)?

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Problem 42

To get to work, a commuter must cross train tracks. The time the train arrives varies slightly from day to day, but the commuter estimates he’ll get stopped on about 15% of work days. During a certain 5-day work week, what is the probability that he
a) gets stopped on Monday and again on Tuesday?
b) gets stopped for the first time on Thursday?
c) gets stopped every day?
d) gets stopped at least once during the week?

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Problem 43

Suppose that in your city 37% of the voters are registered as Democrats, 29% as Republicans, and 11% as members of other parties (Liberal, Right to Life, Green, etc.). Voters not aligned with any official party are termed “Independent.” You are conducting a poll by calling registered voters at random. In your first three calls, what is the probability you talk to
a) all Republicans?
b) no Democrats?
c) at least one Independent?

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Problem 44

Census reports for a city indicate that 62% of residents classify themselves as Christian, 12% as Jewish, and 16% as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as nonreligious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach
a) all Christians?
b) no Jews?
c) at least one person who is nonreligious?

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Problem 45

You bought a new set of four tires from a manufacturer who just announced a recall because 2% of those tires are defective. What is the probability that at least one of yours is defective?

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Problem 46

For a sales promotion, the manufacturer places winning symbols under the caps of 10% of all Pepsi bottles. You buy a six-pack. What is the probability that you win something?

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Problem 47

On September 11, 2002, the first anniversary of the terrorist attack on the World Trade Center, the New York State Lottery’s daily number came up 9–1–1. An interesting coincidence or a cosmic sign?
a) What is the probability that the winning three numbers match the date on any given day?
b) What is the probability that a whole year passes without this happening?
c) What is the probability that the date and winning lottery number match at least once during any year?
d) If every one of the 50 states has a three-digit lottery, what is the probability that at least one of them will come up 9–1–1 on September 11?

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Problem 48

You shuffle a deck of cards and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, “The next one is due to be black!”
a) Are you correct in thinking that there’s a higher probability that the next card will be black than red? Explain.
b) Is this an example of the Law of Large Numbers? Explain.

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