For Exercise 4.89, see page 266; for Exercise 4.90, see page 266; for Exercise 4.91, see page 268; for Exercise 4.92, see page 269; for Exercises 4.93 and 4.94, see page 270; and for Exercise 4.95, see page 273.
4.96 Find and explain some probabilities.
(a) Can we have an event A that has negative probability? Explain your answer.
(b) Suppose P(A) = 0.3 and P(B) = 0.5. Explain what it means for A and B to be disjoint. Assuming that they are disjoint, find the probability that A or B occurs.
(c) Explain in your own words the meaning of the rule P(S) = 1.
(d) Consider an event A. What is the name for the event that A does not occur? If P(A) = 0.4, what is the probability that A does not occur?
(e) Suppose that A and B are independent and that P(A) = 0.8 and P(B) = 0.3. Explain the meaning of the event {A and B}, and find its probability.
4.97 Unions.
(a) Assume that P(A) = 0.2, P(B) = 0.4, and P(C) = 0.1. If the events A, B, and C are disjoint, find the probability that the union of these events occurs.
(b) Draw a Venn diagram to illustrate your answer to part (a).
(c) Find the probability of the complement of the union of A, B, and C.
4.98 Conditional probabilities. Suppose that P(A) = 0.4, P(B) = 0.3, and P(B | A) = 0.4.
(a) Find the probability that both A and B occur.
(b) Use a Venn diagram to explain your calculation.
(c) What is the probability of the event that B occurs and A does not?
4.99 Find the probabilities. Suppose that the probability that A occurs is 0.5 and the probability that A and B occur is 0.2.
(a) Find the probability that B occurs given that A occurs.
(b) Illustrate your calculations in part (a) using a Venn diagram.
4.100 Why not? Suppose that P(B) = 0.6. Explain why P(A and B) cannot be 0.7.
4.101 Is the calcium intake adequate? In the population of young children eligible to participate in a study of whether or not their calcium intake is adequate, 52% are 5 to 10 years of age and 48% are 11 to 13 years of age. For those who are 5 to 10 years of age, 18% have inadequate calcium intake. For those who are 11 to 13 years of age, 57% have inadequate calcium intake.18
(a) Use letters to define the events of interest in this exercise.
(b) Convert the percents given to probabilities of the events you have defined.
(c) Use a tree diagram similar to Figure 4.19 (page 272) to calculate the probability that a randomly selected child from this population has an inadequate intake of calcium.
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4.102 Use Bayes’s rule. Refer to the previous exercise. Use Bayes’s rule to find the probability that a child from this population who has inadequate intake is 11 to 13 years old.
4.103 Are the events independent? Refer to the previous two exercises. Are the age of the child and whether or not the child has adequate calcium intake independent? Calculate the probabilities that you need to answer this question and write a short summary of your conclusion.
4.104 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.
(a) P(A or B) is always equal to the sum of P(A) and P(B).
(b) The probability of an event minus the probability of its complement is always equal to 1.
(c) Two events are disjoint if P(B | A) = P(B).
4.105 Exercise and sleep. Suppose that 42% of adults get enough sleep, 39% get enough exercise, and 28% do both. Find the probabilities of the following events:
(a) Enough sleep and not enough exercise.
(b) Not enough sleep and enough exercise.
(c) Not enough sleep and not enough exercise.
(d) For each of parts (a), (b), and (c), state the rule that you used to find your answer.
4.106 Exercise and sleep. Refer to the previous exercise. Draw a Venn diagram showing the probabilities for exercise and sleep.
4.107 Lying to a teacher. Suppose that 53% of high school students would admit to lying at least once to a teacher during the past year and that 24% of students are male and would admit to lying at least once to a teacher during the past year.19 Assume that 44% of the students are male. What is the probability that a randomly selected student is either male or would admit to lying to a teacher during the past year? Be sure to show your work and indicate all the rules that you use to find your answer.
4.108 Lying to a teacher. Refer to the previous exercise. Suppose that you select a student from the subpopulation of those who would admit to lying to a teacher during the past year. What is the probability that the student is female? Be sure to show your work and indicate all the rules that you use to find your answer.
4.109 Attendance at two-year and four-year colleges. In a large national population of college students, 61% attend four-year institutions and the rest attend two-year institutions. Males make up 44% of the students in the four-year institutions and 41% of the students in the two-year institutions.
(a) Find the four probabilities for each combination of gender and type of institution in the following table. Be sure that your probabilities sum to 1.
| Men | Women | |
|---|---|---|
| Four-year institution | ||
| Two-year institution |
(b) Consider randomly selecting a female student from this population. What is the probability that she attends a four-year institution?
4.110 Draw a tree diagram. Refer to the previous exercise. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the type of institution attended and then identify the gender of the student.
4.111 Draw a different tree diagram for the same setting. Refer to the previous two exercises. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the gender of the student and then identify the type of institution attended. Explain why the probabilities in this tree diagram are different from those that you used in the previous exercise.
4.112 Education and income. Call a household prosperous if its income exceeds $100,000. Call the household educated if the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated. According to the Current Population Survey, P(A) = 0.138, P(B) = 0.261, and the probability that a household is both prosperous and educated is P(A and B) = 0.082. What is the probability P(A or B) that the household selected is either prosperous or educated?
4.113 Find a conditional probability. In the setting of the previous exercise, what is the conditional probability that a household is prosperous, given that it is educated? Explain why your result shows that events A and B are not independent.
4.114 Draw a Venn diagram. Draw a Venn diagram that shows the relation between the events A and B in Exercise 4.112. Indicate each of the following events on your diagram and use the information in Exercise 4.112 to calculate the probability of each event. Finally, describe in words what each event is.
(a) {A and B}.
(b) {Ac and B}.
(c) {A and Bc}.
(d) {Ac and Bc}.
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4.115 Sales of cars and light trucks. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. In a recent year, 69% of vehicles sold were light trucks, 78% were domestic, and 55% were domestic light trucks. Let A be the event that a vehicle is a car and B the event that it is imported. Write each of the following events in set notation and give its probability.
(a) The vehicle is a light truck.
(b) The vehicle is an imported car.
4.116 Job offers. Emily is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let A, B, and C be the events where Emily is offered a job by
A = the Connecticut Office of the Chief Medical Examiner
B = the New Jersey Division of Criminal Justice
C = the federal Disaster Mortuary Operations Response Team
Julie writes down her personal probabilities for being offered these jobs:
P(A) = 0.6 P(B) = 0.5 P(C) = 0.3
P(A and B) = 0.3 P(A and C) = 0.1 P(B and C) = 0.1
P(A and B and C) = 0
Make a Venn diagram of the events A, B, and C. As in Figure 4.18 (page 267), mark the probabilities of every intersection involving these events and their complements. Use this diagram for Exercises 4.117, 4.118, and 4.119.
4.117 Find the probability of at least one offer. What is the probability that Julie is offered at least one of the three jobs?
4.118 Find the probability of another event. What is the probability that Julie is offered both the Connecticut and New Jersey jobs, but not the federal job?
4.119 Find a conditional probability. If Julie is offered the federal job, what is the conditional probability that she is also offered the New Jersey job? If Julie is offered the New Jersey job, what is the conditional probability that she is also offered the federal job?
4.120 Conditional probabilities and independence. Using the information in Exercise 4.115, answer these questions.
(a) Given that a vehicle is imported, what is the conditional probability that it is a light truck?
(b) Are the events “vehicle is a light truck” and “vehicle is imported” independent? Justify your answer.
Genetic counseling. Conditional probabilities and Bayes’s rule are a basis for counseling people who may have genetic defects that can be passed to their children. Exercises 4.121, 4.112, and 4.123 concern genetic counseling settings.
4.121 Albinism. People with albinism have little pigment in their skin, hair, and eyes. The gene that governs albinism has two forms (called alleles), which we denote by a and A. Each person has a pair of these genes, one inherited from each parent. A child inherits one of each parent’s two alleles independently with probability 0.5. Albinism is a recessive trait, so a person is albino only if the inherited pair is aa.
(a) Beth’s parents are not albino but she has an albino brother. This implies that both of Beth’s parents have type Aa. Why?
(b) Which of the types aa, Aa, AA could a child of Beth’s parents have? What is the probability of each type?
(c) Beth is not albino. What are the conditional probabilities for Beth’s possible genetic types, given this fact? (Use the definition of conditional probability.)
4.122 Find some conditional probabilities. Beth knows the probabilities for her genetic types from part (c) of the previous exercise. She marries Bob, who is albino. Bob’s genetic type must be aa.
(a) What is the conditional probability that a child of Beth and Bob is non-albino if Beth has type Aa? What is the conditional probability of a non-albino child if Beth has type AA?
(b) Beth and Bob’s first child is non-albino. What is the conditional probability that Beth is a carrier, type Aa?
4.123 Muscular dystrophy. Muscular dystrophy is an incurable muscle-wasting disease. The most common and serious type, called DMD, is caused by a sex-linked recessive mutation. Specifically, women can be carriers but do not get the disease; a son of a carrier has probability 0.5 of having DMD; a daughter has probability 0.5 of being a carrier. As many as one-third of DMD cases, however, are due to spontaneous mutations in sons of mothers who are not carriers. Toni has one son, who has DMD.
In the absence of other information, the probability is 1/3 that the son is the victim of a spontaneous mutation and 2/3 that Toni is a carrier. There is a screening test called the CK test that is positive with probability 0.7 if a woman is a carrier and with probability 0.1 if she is not. Toni’s CK test is positive. What is the probability that she is a carrier?