## SECTION 4.2 Exercises

For Exercises 4.14 and 4.15, see pages 180–181; for 4.16 to 4.20, see pages 183–184; for 4.21 to 4.23, see pages 185–186; for 4.24, see page 187; for 4.25 and 4.26, see page 189; and for 4.27 to 4.29, see page 190.

### Question

4.30 Support for casino in Toronto.

In an effort to seek the public's input on the establishment of a casino, Toronto's city council enlisted an independent analytics research company to conduct a public survey. A random sample of 902 adult Toronto residents were asked if they support the casino in Toronto.8 Here are the results:

 Response Stronglysupport Somewhatsupport Mixedfeelings Probability 0.16 0.26 ? Response Somewhatoppose Stronglyoppose Don'tknow Probability 0.14 0.36 0.01
1. What probability should replace “?” in the distribution?
2. What is the probability that a randomly chosen adult Toronto resident supports (strongly or somewhat) a casino?

### Question

4.31 Confidence in institutions.

A Gallup Poll (June 1–4, 2013) interviewed a random sample of 1529 adults (18 years or older). The people in the sample were asked about their level of confidence in a variety of institutions in the United States. Here are the results for small and big businesses:9

 Greatdeal Quitea lot Some Verylittle None Noopinion Smallbusiness 0.29 0.36 0.27 0.07 0.00 0.01 Bigbusiness 0.09 0.13 0.43 0.31 0.02 0.02
1. What is the probability that a randomly chosen person has either no opinion, no confidence, or very little confidence in small businesses? Find the similar probability for big businesses.
2. Using your answer from part (a), determine the probability that a randomly chosen person has at least some confidence in small businesses. Again based on part (a), find the similar probability for big businesses.

### Question

4.32 Demographics—language.

Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asian/Pacific region:10

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 Language English French Sino-Tibetan Other Probability 0.581 0.217 0.033 ?

1. What probability should replace “?” in the distribution?
2. Only English and French are considered official languages. What is the probability that a randomly chosen Canadian's mother tongue is not an official language?

### Question

4.33 Online health information.

Based on a random sample of 1066 adults (18 years or older), a Harris Poll (July 13–18, 2010) estimates that 175 million U.S. adults have gone online for health information. Such individuals have been labeled as “cyberchondriacs.” Cyberchondriacs in the sample were asked about the success of their online search for information about health topics. Here is the distribution of responses:11

 Verysuccessful Somewhatsuccessful Neither successfulnor unsuccessful Probability 0.41 0.45 0.04 Somewhatunsuccessful Veryunsuccessful Declineto answer Probability 0.05 0.03 0.02
1. Show that this is a legitimate probability distribution.
2. What is the probability that a randomly chosen cyberchondriac feels that his or her search for health information was somewhat or very successful?

### Question

4.34 World Internet usage.

Approximately 40.4% of the world's population uses the Internet (as of July 2014).12 Furthermore, a randomly chosen Internet user has the following probabilities of being from the given country of the world:

 Region China U.S. India Japan Probability 0.2197 0.0958 0.0833 0.0374
1. What is the probability that a randomly chosen Internet user does not live in one of the four countries listed in this table?
2. What is the probability that a randomly chosen Internet user does not live in the United States?
3. At least what proportion of Internet users are from Asia?

### Question

4.35 Modes of transportation.

Governments (local and national) find it important to gather data on modes of transportation for commercial and workplace movement. Such information is useful for policymaking as it pertains to infrastructure (like roads and railways), urban development, energy use, and pollution. Based on 2011 Canadian and 2012 U.S. government data, here are the distributions of the primary means of transportation to work for employees working outside the home:13

 Car (selfor pool) Publictransportation Bicycle ormotorcycle Walk Other Canada ? 0.120 0.013 0.057 0.014 U.S. ? 0.052 0.006 0.029 0.013
1. What is the probability that a randomly chosen Canadian employee who works outside the home uses an automobile? What is the probability that a randomly chosen U.S. employee who works outside the home uses an automobile?
2. Transportation systems primarily based on the automobile are regarded as unsustainable because of the excessive energy consumption and the effects on the health of populations. The Canadian government includes public transit, walking, and cycles as “sustainable” modes of transportation. For both countries, determine the probability that a randomly chosen employee who works outside home uses sustainable transportation. How do you assess the relative status of sustainable transportation for these two countries?

### Question

4.36 Car colors.

Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for cars purchased in South America in 2012:14

 Color Silver White Black Gray Red Brown Probability 0.29 0.21 0.19 0.13 0.09 0.05
1. What is the probability that a randomly chosen car is either silver or white?
2. In North America, the probability of a new car being blue is 0.07. What can you say about the probability of a new car in South America being blue?

### Question

4.37 Land in Iowa.

Choose an acre of land in Iowa at random. The probability is 0.92 that it is farmland and 0.01 that it is forest.

1. What is the probability that the acre chosen is not farmland?
2. What is the probability that it is either farmland or forest?
3. What is the probability that a randomly chosen acre in Iowa is something other than farmland or forest?

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### Question

4.38 Stock market movements.

You watch the price of the Dow Jones Industrial Index for four days. Give a sample space for each of the following random phenomena.

1. You record the sequence of up-days and down-days.
2. You record the number of up-days.

### Question

4.39 Colors of M&M'S.

The colors of candies such as M&M'S are carefully chosen to match consumer preferences. The color of an M&M drawn at random from a bag has a probability distribution determined by the proportions of colors among all M&M'S of that type.

1. Here is the distribution for plain M&M'S:
 Color Blue Orange Green Brown Yellow Red Probability 0.24 0.20 0.16 0.14 0.14 ?
What must be the probability of drawing a red candy?
2. What is the probability that a plain M&M is any of orange, green, or yellow?

### Question

4.40 Almond M&M'S.

Exercise 4.39 gives the probabilities that an M&M candy is each of blue, orange, green, brown, yellow, and red. If “Almond” M&M'S are equally likely to be any of these colors, what is the probability of drawing a blue Almond M&M?

### Question

4.41 Legitimate probabilities?

In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate—that is, satisfies the rules of probability. If not, give specific reasons for your answer.

1. When a coin is spun, and .
2. When a coin flipped twice, , , , and .
3. Plain M&M'S have not always had the mixture of colors given in Exercise 4.39. In the past there were no red candies and no blue candies. Tan had probability 0.10, and the other four colors had the same probabilities that are given in Exercise 4.39.

### Question

4.42 Who goes to Paris?

Abby, Deborah, Sam, Tonya, and Roberto work in a firm's public relations office. Their employer must choose two of them to attend a conference in Paris. To avoid unfairness, the choice will be made by drawing two names from a hat. (This is an SRS of size 2.)

1. Write down all possible choices of two of the five names. This is the sample space.
2. The random drawing makes all choices equally likely. What is the probability of each choice?
3. What is the probability that Tonya is chosen?
4. What is the probability that neither of the two men (Sam and Roberto) is chosen?

### Question

4.43 Equally likely events.

For each of the following situations, explain why you think that the events are equally likely or not.

1. The outcome of the next tennis match for Victoria Azarenka is either a win or a loss. (You might want to check the Internet for information about this tennis player.)
2. You draw a king or a two from a shuffled deck of 52 cards.
3. You are observing turns at an intersection. You classify each turn as a right turn or a left turn.
4. For college basketball games, you record the times that the home team wins and the number of times that the home team loses.

### Question

4.44 Using Internet sources.

Internet sites often vanish or move, so references to them can't be followed. In fact, 13% of Internet sites referenced in major scientific journals are lost within two years after publication.

1. If a paper contains seven Internet references, what is the probability that all seven are still good two years later?
2. What specific assumptions did you make in order to calculate this probability?

### Question

4.45 Everyone gets audited.

Wallen Accounting Services specializes in tax preparation for individual tax returns. Data collected from past records reveals that 9% of the returns prepared by Wallen have been selected for audit by the Internal Revenue Service. Today, Wallen has six new customers. Assume the chances of these six customers being audited are independent.

1. What is the probability that all six new customers will be selected for audit?
2. What is the probability that none of the six new customers will be selected for audit?
3. What is the probability that exactly one of the six new customers will be selected for audit?

### Question

4.46 Hiring strategy.

A chief executive officer (CEO) has resources to hire one vice president or three managers. He believes that he has probability 0.6 of successfully recruiting the vice president candidate and probability 0.8 of successfully recruiting each of the manager candidates. The three candidates for manager will make their decisions independently of each other. The CEO must successfully recruit either the vice president or all three managers to consider his hiring strategy a success. Which strategy should he choose?

### Question

4.47 A random walk on Wall Street?

The “random walk” theory of securities prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year and that the probability that our portfolio rises in price in any one year is 0.65. (This probability is approximately correct for a portfolio containing equal dollar amounts of all common stocks listed on the New York Stock Exchange.)

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1. What is the probability that our portfolio goes up for three consecutive years?
2. If you know that the portfolio has risen in price two years in a row, what probability do you assign to the event that it will go down next year?
3. What is the probability that the portfolio's value moves in the same direction in both of the next two years?

### Question

4.48 The multiplication rule for independent events.

The probability that a randomly selected person prefers the vehicle color white is 0.24. Can you apply the multiplication rule for independent events in the situations described in parts (a) and (b)? If your answer is Yes, apply the rule.

1. Two people are chosen at random from the population. What is the probability that both prefer white?
2. Two people who are sisters are chosen. What is the probability that both prefer white?
3. Write a short summary about the multiplication rule for independent events using your answers to parts (a) and (b) to illustrate the basic idea.

### Question

4.49 What's wrong?

In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

1. If two events are disjoint, we can multiply their probabilities to determine the probability that they will both occur.
2. If the probability of is 0.6 and the probability of is 0.5, the probability of both and happening is 1.1.
3. If the probability of is 0.35, then the probability of the complement of is −0.35.

### Question

4.50 What's wrong?

In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

1. If the sample space consists of two outcomes, then each outcome has probability 0.5.
2. If we select a digit at random, then the probability of selecting a 2 is 0.2.
3. If the probability of is 0.2, the probability of is 0.3, and the probability of and is 0.5, then and are independent.

### Question

4.51 Playing the lottery.

An instant lottery game gives you probability 0.02 of winning on any one play. Plays are independent of each other. If you play five times, what is the probability that you win at least once?

### Question

4.52 Axioms of probability.

Show that any assignment of probabilities to events that obeys Rules 2 and 3 on page 182 automatically obeys the complement rule (Rule 4). This implies that a mathematical treatment of probability can start from just Rules 1, 2, and 3. These rules are sometimes called axioms of probability.

### Question

4.53 Independence of complements.

Show that if events and obey the multiplication rule, , then and the complement of also obey the multiplication rule, . That is, if events and are independent, then and are also independent. (Hint: Start by drawing a Venn diagram and noticing that the events “ and ” and “ and ” are disjoint.)