Solved: Sample spaces. For each of the following, list the

Sample spaces. For each of the following, list the sam- ple 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.

Sample spaces. For each of the following, list the sam- ple 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.Sample spaces. 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 childs 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.

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

Rain. 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?

Winter. 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. Im 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.)

Snow. After an unusually dry autumn, a radio an- nouncer is heard to say, Watch out! Well pay for these sunny days later on this winter. Explain what hes trying to say, and comment on the validity of his reasoning.

Cold streak. 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.

Crash. 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?

Fire insurance. Insurance companies collect annual payments from homeowners in exchange for paying to rebuild houses that burn down. a) Why should you be reluctant to accept a $300 pay- ment from your neighbor to replace his house should it burn down during the coming year? b) Why can the insurance company make that offer?

Jackpot. On January 20, 2000, the International Gaming Technology company issued a press release: (LAS VEGAS, Nev.)Cynthia Jay was smiling ear to ear as she walked into the news conference at The Desert Inn Resort in Las Vegas today, and well she should. Last night, the 37-year-old cocktail waitress won the worlds largest slot jackpot$34,959,458on a Megabucks machine. She said she had played $27 in the machine when the jackpot hit. Nevada Megabucks has produced 49 major winners in its 14-year history. The top jackpot builds from a base amount of $7 million and can be won with a 3-coin ($3) bet. a) How can the Desert Inn afford to give away millions of dollars on a $3 bet? b) Why did the company issue a press release? Wouldnt most businesses want to keep such a huge loss quiet?

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

Scratch off. 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?

Vehicles. Suppose that 46% of families living in a certain county own a car and 18% own an SUV. The Addition Rule might suggest, then, that 64% of families own either a car or an SUV. Whats wrong with that reasoning?

Homes. Funding for many schools comes from taxes based on assessed values of local properties. Peoples 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 resi- dences have a garage or a pool. Whats wrong with that reasoning?

Speeders. Traffic checks on a certain section of high- way suggest that 60% of drivers are speeding there. Since , the Multiplication Rule might suggest that theres a 36% chance that two vehicles in a row are both speeding. Whats wrong with that reasoning?

Lefties. Although its hard to be definitive in clas- sifying people as right- or left-handed, some studies sug- gest that about 14% of people are left-handed. Since , the Multiplication Rule might sug- gest that theres about a 2% chance that a brother and a sister are both lefties. Whats wrong with that reasoning?

College admissions. For high school students gradu- ating in 2007, college admissions to the nations 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 appli- cants, Stanford 10%, and Penn 16%. Jorge has applied to all three. Assuming that hes a typical applicant, he fig- ures 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?

College admissions II. In Exercise 17, 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?

Car repairs. 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?

Stats projects. In a large Introductory Statistics lecture hall, the professor reports that 55% of the students en- rolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. What is the probability that the first groupmate you meet has studied a) two or more semesters of Calculus? b) some Calculus? c) no more than one semester of Calculus?

More repairs. Consider again the auto repair rates de- scribed in Exercise 19. 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?

Another project. You are assigned to be part of a group of three students from the Intro Stats class described in Exercise 20. What is the probability that of your other two groupmates, a) neither has studied Calculus? b) both have studied at least one semester of Calculus? c) at least one has had more than one semester of Calculus?

Repairs, again. You used the Multiplication Rule to calculate repair probabilities for your cars in Exercise 21. a) What must be true about your cars in order to make that approach valid? b) Do you think this assumption is reasonable? Explain.

Final project. You used the Multiplication Rule to calculate probabilities about the Calculus background of your Statistics groupmates in Exercise 22. a) What must be true about the groups in order to make that approach valid? b) Do you think this assumption is reasonable? Explain.

Energy 2007. A Gallup poll in March 2007 asked 1005 U.S. adults whether increasing domestic energy produc- tion 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 1005 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?

Failing fathers? A Pew Research poll in 2007 asked 2020 U.S. adults whether fathers today were doing as good a job of fathering as fathers of 2030 years ago. Heres how they responded: If we select a respondent at random from this sample of 2020 adults, a) what is the probability that the selected person re- sponded Worse? b) what is the probability that the person responded the Same or Better?

More energy. Exercise 25 shows the results of a Gallup Poll about energy. Suppose we select three people at ran- dom from this sample. a) What is the probability that all three responded Pro- tect 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.

Fathers revisited. Consider again the results of the poll about fathering discussed in Exercise 26. If we select two people at random from this sample, a) what is the probability that both think fathers are bet- ter today? b) what is the probability that neither thinks fathers are better today? c) what is the probability that one person thinks fathers are better today and the other doesnt? d) What assumption did you make in computing these probabilities? e) Explain why you think that assumption is reasonable.

Polling. As mentioned in the chapter, opinion-polling organizations contact their respondents by sampling ran- dom telephone numbers. Although interviewers now can reach about 76% of U.S. households, the percentage of those contacted who agree to cooperate with the survey has fallen from 58% in 1997 to only 38% in 2003 (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 is the probability (in 2003) of failing to contact a household or of contacting the household but not get- ting them to agree to the interview? c) Show another way to calculate the probability in part b.

Polling, part II. According to Pew Research, the con- tact rate (probability of contacting a selected household) was 69% in 1997 and 76% in 2003. However, the coopera- tion rate (probability of someone at the contacted house- hold agreeing to be interviewed) was 58% in 1997 and dropped to 38% in 2003. a) What is the probability (in 2003) of obtaining an in- terview 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 ran- domly selected household in 1997 or in 2003?

M&Ms. The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&Ms, 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 is the probabil- ity 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&Ms in a row, what is the proba- bility that 1) they are all brown? 2) the third one is the first one thats red? 3) none are yellow? 4) at least one is green?

Blood. 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?

Disjoint or independent? In Exercise 31 you calcu- lated probabilities of getting various M&Ms. Some of your answers depended on the assumption that the out- comes 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 doesnt affect the probability of the other. Do you understand the difference between dis- joint 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&Ms 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.

Disjoint or independent? In Exercise 32 you calcu- lated probabilities involving various blood types. Some of your answers depended on the assumption that the out- comes 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 doesnt affect the probability of the other. Do you understand the difference between dis- joint 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, independ- ent, or neither? c) Can disjoint events ever be independent? Explain.

Dice. You roll a fair die three times. What is the proba- bility that a) you roll all 6s? 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 5s?

Slot machine. 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)?

Champion bowler. A certain bowler can bowl a strike 70% of the time. Whats 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)?

The train. 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 hell 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?

Voters. 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 reg- istered 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?

Religion. 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 them- selves 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?

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

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

9/11? On September 11, 2002, the first anniversary of the terrorist attack on the World Trade Center, the New York State Lotterys daily number came up 911. 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 lot- tery 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 911 on September 11?

Red cards. 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 theres 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.