18.8 Moving up. A sociologist studying social mobility in Denmark finds that the probability that the son of a lower-class father remains in the lower class is 0.46. What is the probability that the son moves to one of the higher classes?

18.9 Causes of death. Government data assign a single cause for each death that occurs in the United States. Data from 2013 show that the probability is 0.31 that a randomly chosen death was due to heart disease and 0.23 that it was due to cancer. What is the probability that a death was due either to heart disease or to cancer? What is the probability that the death was due to some other cause?

18.10 Land in Canada. Choose an acre of land in Canada at random. The probability is 0.45 that it is forest and 0.03 that it is pasture.

18.11 Our favorite president. A Gallup Poll on Presidents’ Day 2011 interviewed a random sample of 1015 adult Americans. Those in the sample were asked who they regarded as the greatest U.S. president. Here are the results:

These proportions are probabilities for the random phenomenon of choosing an adult American at random and asking her or his opinion.

18.12 Rolling a die. Figure 18.5 displays several assignments of probabilities to the six faces of a die. We can learn which assignment is actually correct for a particular die only by rolling the die many times. However, some of the assignments are not legitimate assignments of probability. That is, they do not obey the rules. Which are legitimate and which are not? In the case of the illegitimate models, explain what is wrong.

18.13 The demands of college. Select a first-year college student at random and ask how difficult it was for him or her to adjust to the academic demands of college. Here are the probabilities, based on proportions from a large sample survey in 2010 of first-year students:

18.14 Tetrahedral dice. Psychologists sometimes use tetrahedral dice to study our intuition about chance behavior. A tetrahedron (Figure 18.6) is a pyramid with four faces, each a triangle with all sides equal in length. Label the four faces of a tetrahedral die with one, two, three, and four spots. Give a probability model for rolling such a die and recording the number of spots on the down-face. Explain why you think your model is at least close to correct.

18.15 Birth order. A couple plan to have three children. There are eight possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a boy. All eight arrangements are (approximately) equally likely.

18.16 More tetrahedral dice. Tetrahedral dice are described in Exercise 18.14. Give a probability model for rolling two such dice. That is, write down all possible outcomes and give a probability to each. (Example 2 and Figure 18.1 may help you.) What is the probability that the sum of the down-faces is 5?

18.17 Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and, at the same time, rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.

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18.18 Colors of M&M’s. If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made.

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

18.20 Immigration. Suppose that 34% of all adult Americans think that the level of immigration to the United States should be decreased. An opinion poll interviews 1020 randomly chosen Americans and records the sample proportion who feel that the level of immigration to this country should be decreased. This statistic will vary from sample to sample if the poll is repeated. The sampling distribution is approximately Normal with mean 0.34 and standard deviation about 0.015. Sketch this Normal curve and use it to answer the following questions.

18.21 Airplane safety. Suppose that 73% of all adults think that the overall safety of airline travel is either good or excellent. An opinion poll plans to ask an SRS of 1097 adults about airplane safety. The proportion of the sample who think that the overall safety of airline travel is either good or excellent will vary if we take many samples from this same population. The sampling distribution of the sample proportion is approximately Normal with mean 0.73 and standard deviation about 0.013. Sketch this Normal curve and use it to answer the following questions.

18.22 Immigration (optional). In the setting of Exercise 18.20, what is the probability of getting a sample in which less than 30% of those sampled think that the level of immigration to this country should be decreased? (Use Table B.)

18.23 Airplane safety (optional). In the setting of Exercise 18.21, what is the probability of getting a sample in which more than 75% think that the overall safety of airline travel is either good or excellent? (Use Table B.)

18.24 Do you jog? An opinion poll asks an SRS of 1500 adults, “Do you happen to jog?” Suppose (as is approximately correct) that the population proportion who jog is p = 0.15. In a large number of samples, the proportion who answer Yes will be approximately Normally distributed with mean 0.15 and standard deviation 0.009. Sketch this Normal curve and use it to answer the following questions.

18.25 Applying to college. You ask an SRS of 1500 college students whether they applied for admission to any other college. Suppose that, in fact, 35% of all college students applied to colleges besides the one they are attending. (That’s close to the truth.) The sampling distribution of the proportion of your sample who say Yes is approximately Normal with mean 0.35 and standard deviation 0.01. Sketch this Normal curve and use it to answer the following questions.

18.26 Generating a sampling distribution. Let us illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam:

The parameter of interest is the mean score in this population. The sample is an SRS of size drawn from the population. Because the students are labeled 0 to 9, a single random digit from Table A chooses one student for the sample.

18.27 Odds and personal probability. One way to determine your personal probability about an event is to ask what you would consider a fair bet that the event will occur. Suppose in August 2015 you believed it fair that a bet of $2 should win $7 if the Green Bay Packers win Super Bowl 50.

18.28 The addition rule. Probability rule D states: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. This is sometimes called the addition rule for disjoint events. A more general form of the addition rule is: the probability that one or the other of two events occurs is the sum of their individual probabilities minus the probability that both occur. To verify this rule, suppose you roll two casino dice as in Example 2. Refer to the outcomes in Figure 18.1 to answer the following.

Note: The outcomes you were asked to count in part (c) are among those counted in parts (a) and (b). When we combine the outcomes in parts (a) and (b), we “double count” the outcomes in part (c). One of these “double counts” must be eliminated so that the combination corresponds to the outcomes in part (d). This is the reason that, in the addition rule, you subtract the probability that both occur from the sum of their individual probabilities.