Chapter 14

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?

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?

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?

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?

In a large Introductory Statistics lecture hall, the professor reports that \(55 \%\) of the students enrolled 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?

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.

As mentioned in the chapter, opinion-polling organizations contact their respondents by sampling random 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 getting them to agree to the interview? c) Show another way to calculate the probability in part b.

According to Pew Research, the contact rate (probability of contacting a selected household) was \(69 \%\) in 1997 and \(76 \%\) in 2003 . However, the cooperation rate (probability of someone at the contacted household agreeing to be interviewed) was \(58 \%\) in 1997 and dropped to \(38 \%\) in 2003 . a) What is the probability (in 2003) 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 \(2003 ?\)

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 is the probability 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 probability 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?

The American Red Cross says that about \(45 \%\) of the U.S. population has Type O blood, \(40 \%\) Type A, \(11 \%\) Type \(\mathrm{B}\), and the rest Type \(\mathrm{AB}\). 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 \(\mathrm{O}\) ? b) Among four potential donors, what is the probability that 1) all are Type \(\mathrm{O}\) ? 2) no one is Type \(\mathrm{AB}\) ? 3) they are not all Type \(\mathrm{A}\) ? 4) at least one person is Type B?