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:

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

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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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

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:

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

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

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.

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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.

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.

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?

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.

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.)

4.43 Equally likely events.

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

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.

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.

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?

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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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.

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.

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.

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?

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.

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.)