7.9: Practice and Exploration

1. Suppose that six equally qualified employees are eligible for a promotion. Four of these employees are Asian American. Two employees are to be selected at random and neither are Asian American. Make a table like Table 7.1 that lists all the possible selections of two employees from the pool of six employees. Use the labels \(A_1\), \(A_2\), \(A_3\), and \(A_4\) to denote the Asian American employees. Use the labels \(N_1\) and \(N_2\) for the two employees who are not Asian American. You should get fifteen such choices. Remember that the order in which the employees is selected does not matter.
2. Determine how many of these possible selections correspond to choosing the two individuals who are not Asian American. Use this number to compute the probability of choosing the two individuals who are not Asian American.
3. In an employment discrimination case, a lawyer is presenting evidence that their client, in this case the company, has a racially unbiased method for evaluating and hiring employees. As part of this evidence the lawyer has had a statistician compute the probability that no Hispanic applicants would be hired from a specified applicant pool, under the assumption that there is no bias. The lawyer states that this probability is equal to 0.001, saying that this is not a rare occurrence. Using your intuitive ideas about probability, do you agree with the lawyer that this would not be a rare occurrence?

4. A ballot box contains one hundred ballots, seventy-five for Candidate A and twenty-five for Candidate B. Suppose that a ballot is selected at random in a way that each ballot of an equal chance of being selected. What is the probability that the selected ballot is for Candidate A? What are the odds that the selected ballot is for Candidate A? What is the odds ratio of selecting a ballot for Candidate A compared to selecting a Ballot for Candidate B?

5. The odds that a senior at a certain suburban high school will go to college is 2 to 1 while the odds that a senior at a certain urban high school will go to college is 1 to 3. Compute the probabilities corresponding to each of these odds. According to these odds, is it more likely that a high school senior will go to college from the urban or the suburban high school? What is the odds ratio comparing the suburban high school to the urban high school?

6. A professor at a large university is interested in whether students who use the free department tutoring center are less likely to fail their course. The professor includes an extra credit question on their final exam that asks whether the student included the free tutoring center as part of their usual study habits for the class. The professor then aggregated these responses with the final grades in the course to produce the following table:

Used Tutoring Center?	Passed Course	Failed Course
Yes	24	12
No	33	9

Based on the observed data, compute the odds ratio for passing the course when using the tutoring center compared to not using the tutoring center. Interpret the result that you get. Are you surprised by the result? What confounding factor might explain why you got this result?

7. Suppose that a standard deck of cards is shuffled and the top card is selected. Assuming that each of the cards is equally likely to be on the top of the deck, what is the probability of drawing an ace? Convert this probability to chance and to approximate odds. Next compute the probability of drawing a diamond. Convert this probability to chance and to approximate odds. Compute and interpret the odds ratio for drawing an ace compared to drawing a diamond.
8. The complement rule is a result from probability that states that the probability that an outcome does not occur is one minus the probability that the outcome does occur. This result was used several times in this chapter without naming the result that we were using. The simplest case of applying the complement rule is for a coin flip. In terms of the experiment of flipping a coin, explain in your own words why the probability of flipping heads is equal to one minus the probability of not flipping heads, that is, flipping tails. How would you translate the complement rule from probability to chance?
9. You and a friend are preparing to play a role-playing game. When you unpack your dice, you notice that neither of you brought an eight-sided die. Your friend states that this is no problem because both of you brought a four-sided die, and if you roll both four-sided dice at one time and add the result that this will be equivalent to rolling a single eight-sided die. Do you think that this is true?
10. In the study of sports activities and lower back pain in adolescents, it was reported that the odds ratio for developing lower back pain for students who participated in soccer compared to the no sports group was 1.77, while the odds ratio for developing lower back pain for students who participated in archery compared to the no sports group was 1.10. Based on these two odds ratios, can we conclude that students who participate in soccer are more likely to develop lower back pain than students who participate in archery? Why or why not?
11. Earlier we reported that the odds of suffering a shark-related fatality within your lifetime is 1 in 4,332,817. The same report from the Florida Museum states that the odds of dying in a bicycle accident during your lifetime is 1 in 4,919. Based on these odds, compute the probability of dying in a bicycle accident during your lifetime. What is the odds ratio of dying in a bicycle accident compared to a fatal shark attack?
12. The epic poem De Vetula considered computing the probabilities associated with each of the possible sums observed when rolling three six-sided dice. Suppose that instead of a die, we have a coin where one side is labeled 0 and the other side is labeled 1. Doing the same analysis used in De Vetula, find the probabilities associated with each possible sum of three flips of this coin.