SECTION 4.2 EXERCISES

For Exercise 4.8, see page 222; for Exercise 4.9, see page 222; for Exercises 4.10 and 4.11, see page 225; for Exercises 4.12 and 4.13, see page 226; for Exercise 4.14, see page 228; for Exercise 4.15, see page 229; and for Exercise 4.16, see page 229.

Question 4.17

4.17 What is the sample space? For each of the following questions, define a sample space for the associated random phenomenon. Explain your answers. Be sure to specify units if that is appropriate.

  1. (a) Will it rain tomorrow?

  2. (b) How many times do you tweet in a typical day?

  3. (c) What is the average age of your Facebook friends?

  4. (d) What are the majors for students at your college?

Question 4.18

4.18 Probability rules. For each of the following situations, state the probability rule or rules that you would use and apply it or them. Write a sentence explaining how the situation illustrates the use of the probability rules.

  1. (a) The probability of event A is 0.417. What is the probability that event A does not occur?

  2. (b) A coin is tossed four times. The probability of zero heads is 1/16 and the probability of zero tails is 1/16. What is the probability that all four tosses result in the same outcome?

  3. (c) Refer to part (b). What is the probability that there is at least one head and at least one tail?

  4. (d) The probability of event A is 0.4 and the probability of event B is 0.8. Events A and B are disjoint. Can this happen?

  5. (e) Event A is very rare. Its probability is −0.04. Can this happen?

Question 4.19

4.19 Equally likely events. For each of the following situations, explain why you think that the events are equally likely or not. Explain your answers.

  1. (a) The outcome of the next tennis match for Sloane Stevens is either a win or a loss. (You might want to check the Internet for information about this tennis player.)

  2. (b) You roll a fair die and get a 3 or a 4.

  3. (c) You are observing turns at an intersection. You classify each turn as a right turn or a left turn.

  4. (d) For college basketball games, you record the times that the home team wins and the number of times that the home team loses.

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Question 4.20

4.20 The multiplication rule for independent events. The probability that a randomly selected person prefers the vehicle color white is 0.24. Can you apply the multiplication rule for independent events in the situations described in parts (a) and (b)? If your answer is Yes, apply the rule.

  1. (a) Two people are chosen at random from the population. What is the probability that both prefer white?

  2. (b) Two people who are sisters are chosen. What is the probability that both prefer white?

  3. (c) Write a short summary about the multiplication rule for independent events using your answers to parts (a) and (b) to illustrate the basic idea.

Question 4.21

4.21 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

  1. (a) If two events are disjoint, we can multiply their probabilities to determine the probability that they will both occur.

  2. (b) If the probability of A is 0.7 and the probability of B is 0.5, the probability of both A and B happening is 1.2.

  3. (c) If the probability of A is 0.45, then the probability of the complement of A is −0.45.

Question 4.22

4.22 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

  1. (a) If the sample space consists of two outcomes, then each outcome has probability 0.5.

  2. (b) If we select a digit at random, then the probability of selecting a 3 is 0.3.

  3. (c) If the probability of A is 0.3, the probability of B is 0.4, and the probability of A and B is 0.5, then A and B are independent.

Question 4.23

4.23 Evaluating web page designs. You are a web page designer and you set up a page with four different links. A user of the page can click on one of the links or he or she can leave that page. Describe the sample space for the outcome of someone visiting your web page.

Question 4.24

4.24 Record the length of time spent on the page. Refer to the previous exercise. You also decide to measure the length of time a visitor spends on your page. Give the sample space for this measure.

Question 4.25

4.25 Distribution of blood types. All human blood can be “ABO-typed” as one of O, A, B, or AB, but the distribution of the types varies a bit among groups of people. Here is the distribution of blood types for a randomly chosen person in the United States:7

Blood type A B AB O
U.S. probability 0.42 0.11 ? 0.44

  1. (a) What is the probability of type AB blood in the United States?

  2. (b) Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen person from the United States can donate blood to Maria?

Question 4.26

4.26 Blood types in Ireland. The distribution of blood types in Ireland differs from the U.S. distribution given in the previous exercise:

Blood type A B AB O
Ireland probability 0.35 0.10 0.03 0.52

Choose a person from the United States and a person from Ireland at random, independently of each other. What is the probability that both have type O blood? What is the probability that both have the same blood type?

Question 4.27

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

  1. (a) Choose a college student at random and record gender and enrollment status: P(female full-time) = 0.44, P(female part-time) = 0.56, P(male full-time) = 0.46, P(male part-time) = 0.54.

  2. (b) Deal a card from a shuffled deck: P(clubs) = 16/52, P(diamonds) = 12/52, P(hearts) = 12/52, P(spades) = 12/52.

  3. (c) Roll a die and record the count of spots on the up-face: P(1) = 1/3, P(2) = 0, P(3) = 1/6, P(4) = 1/3, P(5) = 1/6, P(6) = 0.

Question 4.28

4.28 French and English in Canada. Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asian/Pacific region:8

Language English French Asian/Pacific Other
Probability 0.59 ? 0.07 0.11

  1. (a) What probability should replace “?” in the distribution?

  2. (b) What is the probability that a Canadian’s mother tongue is not English? Explain how you computed your answer.

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Question 4.29

4.29 Education levels of young adults. Choose a young adult (age 25 to 34 years) at random. The probability is 0.12 that the person chosen did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor’s degree.

  1. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor’s degree?

  2. (b) What is the probability that a randomly chosen young adult has at least a high school education?

Question 4.30

image 4.30 Loaded dice. There are many ways to produce crooked dice. To load a die so that 6 comes up too often and 1 (which is opposite 6) comes up too seldom, add a bit of lead to the filling of the spot on the 1 face. Because the spot is solid plastic, this works even with transparent dice. If a die is loaded so that 6 comes up with probability 0.24 and the probabilities of the 2, 3, 4, and 5 faces are not affected, what is the assignment of probabilities to the six faces?

Question 4.31

4.31 Rh blood types. Human blood is typed as O, A, B, or AB and also as Rh-positive or Rh-negative. ABO type and Rh-factor type are independent because they are governed by different genes. In the American population, 84% of people are Rh-positive. Use the information about ABO type in Exercise 4.25 to give the probability distribution of blood type (ABO and Rh) for a randomly chosen American.

Question 4.32

4.32 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.

  1. (a) What is the probability that the ball will land in any one slot?

  2. (b) If you bet on “red,” you win if the ball lands in a red slot. What is the probability of winning?

  3. (c) The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains all multiples of 3, that is, 3, 6, 9, . . . , 36. You place a “column bet” that wins if any of these numbers comes up. What is your probability of winning?

Question 4.33

4.33 Winning the lottery. A state lottery’s Pick 3 game asks players to choose a three-digit number, 000 to 999. The state chooses the winning three-digit number at random so that each number has probability 1/1000. You win if the winning number contains the digits in your number, in any order.

  1. (a) Your number is 059. What is your probability of winning?

  2. (b) Your number is 223. What is your probability of winning?

Question 4.34

4.34 PINs. The personal identification numbers (PINs) for automatic teller machines usually consist of four digits. You notice that most of your PINs have at least one 0, and you wonder if the issuers use lots of 0s to make the numbers easy to remember. Suppose that PINs are assigned at random, so that all four-digit numbers are equally likely.

  1. (a) How many possible PINs are there?

  2. (b) What is the probability that a PIN assigned at random has at least one 0?

Question 4.35

4.35 Universal blood donors. People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of O-negative blood. Only 7% of the American population have O-negative blood. If eight people appear at random to give blood, what is the probability that at least one of them is a universal donor?

Question 4.36

image 4.36 Axioms of probability. Show that any assignment of probabilities to events that obeys Rules 2 and 3 on page 224 automatically obeys the complement rule (Rule 4). This implies that a mathematical treatment of probability can start from just Rules 1, 2, and 3. These rules are sometimes called axioms of probability.

Question 4.37

image 4.37 Independence of complements. Show that if events A and B obey the multiplication rule, P(A and B) = P(A)P(B), then A and the complement Bc of B also obey the multiplication rule, P(A and Bc) = P(A)P(Bc). That is, if events A and B are independent, then A and Bc are also independent. (Hint: Start by drawing a Venn diagram and noticing that the events “A and B” and “A and Bc” are disjoint.)

Mendelian inheritance. Some traits of plants and animals depend on inheritance of a single gene. This is called Mendelian inheritance, after Gregor Mendel (1822–1884). Exercises 4.38 through 4.41 are based on the following information about Mendelian inheritance of blood type.

Each of us has an ABO blood type, which describes whether two characteristics, called A and B, are present. Every one of us has two blood type alleles (gene forms), one inherited from our mother and one from our father. Each of these alleles can be A, B, or O. Which two we inherit determines our blood type. Here is a table that shows what our blood type is for each combination of two alleles:

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Alleles inherited Blood type
A and A A
A and B AB
A and O A
B and B B
B and O B
O and O O

We inherit each of a parent’s two alleles with probability 0.5. We inherit independently from our mother and father.

Question 4.38

4.38 Blood types of children. Emily and Michael both have alleles O and O.

  1. (a) What blood types can their children have?

  2. (b) What is the probability that their next child has each of these blood types?

Question 4.39

4.39 Parents with alleles B and O. Andreona and Caleb both have alleles B and O.

  1. (a) What blood types can their children have?

  2. (b) What is the probability that their next child has each of these blood types?

Question 4.40

4.40 Two children. Samantha has alleles B and O. Dylan has alleles A and B. They have two children. What is the probability that both children have blood type A? What is the probability that both children have the same blood type?

Question 4.41

4.41 Three children. Anna has alleles B and O. Nathan has alleles A and O.

  1. (a) What is the probability that a child of these parents has blood type O?

  2. (b) If Anna and Nathan have three children, what is the probability that all three have blood type O? What is the probability that the first child has blood type O and the next two do not?