202 Chapter 5

5Ð128. Find the probability of getting a full house (three 5Ð131. Find the probability of getting any triple-digit cards of one denomination and two of another) when five number, where all the digits are the same, on a lotto that cards are dealt from an ordinary deck. consists of selecting a three-digit number. 5Ð129. A committee of four people is to be formed from 5Ð132. Find the probability of selecting three science six doctors and eight dentists. Find the probability that the books and four math books from eight science books and committee will consist of the following: nine math books. The books are selected at random. a. All dentists 5Ð133. When three dice are rolled, find the probability of b. Two dentists and two doctors getting a sum of 7. c. All doctors d. Three doctors and one dentist 5Ð134. Find the probability of randomly selecting two e. One doctor and three dentists mathematics books and three physics books from four mathematics books and eight physics books from a box. 5Ð130. An insurance sales representative selects three policies to review. The group of policies he can select from 5Ð135. Find the probability that if five different-sized contains eight life policies, five automobile policies, and washers are arranged in a row, they will be arranged in two homeowner’s policies. Find the probability of selecting order of size. the following: 5Ð136. Using the information in Exercise 4Ð69, Chapter 4, a. All life policies find the probability of each poker hand. b. Both homeowner’s policies a. Royal flush c. All automobile policies b. Straight flush d. One of each policy c. Four of a kind e. Two life and one automobile policies

5–6 In this chapter, the basic concepts and rules of probability are explained. The three types of probability are classical, empirical, and subjective. Classical probability uses sample Summary spaces. Empirical probability uses frequency distributions and is based on observation. In subjective probability, the researcher makes an educated guess about the chance of an event occurring. A probability event consists of one or more outcomes of a probability experiment. Two events are said to be mutually exclusive if they cannot occur at the same time. Events can also be classified as independent or dependent. If events are independent, whether or not the first event occurs does not affect the probability of the next event oc- curring. If the probability of the second event occurring is changed by the occurrence of the first event, then the events are dependent. The complement of an event is the set of outcomes in the that are not included in the outcomes of the event itself. Complementary events are mutually exclusive. Probability problems can be solved by using the addition rules, the multiplication rules, and the complementary event rules. Finally, when the number of outcomes of the sample space is large, probability problems can be solved by using the counting rules in Chapter 4.

Important Terms classical probability 171 empirical mutually exclusive simple event 171 complement of an probability 175 events 182 subjective event 174 equally likely events 171 168 probability 178 compound event 171 event 171 probability 168 Venn diagrams 175 conditional independent events 188 probability probability 191 law of large experiment 168 dependent events 190 numbers 178 sample space 168 Section 5Ð6 Summary 203

Important Formulas

Formula for classical probability: Addition rule 2, for events that are not mutually exclusive: number of P(A or B) P(A) P(B) P(A and B) outcomes Multiplication rule 1, for independent events: in E nE PE total number of nS P(A and B) P(A) • P(B) outcomes in the Multiplication rule 2, for dependent events: sample space P(A and B) P(A) • P(BA) Formula for empirical probability: Formula for : frequency for the class f PA and B PE PBA total frequencies n PA in the distribution Formula for complementary events: — — Addition rule 1, for two mutually exclusive events: PE 1 PE or PE 1 PE — P(A or B) P(A) P(B) or PE PE 1

Review Exercises

5Ð137. When a die is rolled, find the probability of getting b. A yellow or a white sweater a. A5 c. A red, a blue, or a yellow sweater b. A6 d. A sweater that was not white c. A number less than 5 5Ð143. At a swimwear store, the managers found that 5Ð138. When a card is selected from a deck, find the 16 women bought white bathing suits, 4 bought red suits, probability of getting 3 bought blue suits, and 7 bought yellow suits. If a a. A club customer is selected at random, find the probability that b. A face card or a heart she bought the following. c. A 6 and a spade a. A blue suit d. A king b. A yellow or a red suit e. A red card c. A white or a yellow or a blue suit d. A suit that was not red 5Ð139. In a survey conducted at a local restaurant during breakfast hours, 20 people preferred orange juice, 16 5Ð144. When two dice are rolled, find the probability of preferred grapefruit juice, and 9 preferred apple juice with getting breakfast. If a person is selected at random, find the a. A sum of 5 or 6 probability that he or she prefers grapefruit juice. b. A sum greater than 9 c. A sum less than 4 or greater than 9 5Ð140. If a die is rolled one time, find these : d. A sum that is divisible by 4 a. Getting a 5 e. A sum of 14 b. Getting an odd number f. A sum less than 13 c. Getting a number less than 3 5Ð145. The probability that a person owns a car is 0.80, 5Ð141. A recent survey indicated that in a town of 1500 that a person owns a boat is 0.30, and that a person owns households, 850 had cordless telephones. If a household is both a car and a boat is 0.12. Find the probability that a randomly selected, find the probability that it has a cordless person owns either a boat or a car, but not both. telephone. 5Ð146. There is a 0.39 probability that John will purchase 5Ð142. During a sale at a men’s store, 16 white sweaters, 3 a new car, a 0.73 probability that Mary will purchase a red sweaters, 9 blue sweaters, and 7 yellow sweaters were new car, and a 0.36 probability that both will purchase a purchased. If a customer is selected at random, find the new car. Find the probability that neither will purchase probability that he bought the following: a new car. a. A blue sweater 204 Chapter 5 Probability

5Ð147. A Gallup Poll found that 78% of Americans 5Ð155. The probability that Sue will live on campus and worry about the quality and healthfulness of their diet. If buy a new car is 0.37. If the probability that she will live on five people are selected at random, find the probability campus is 0.73, find the probability that she will buy a new that all five worry about the quality and healthfulness of car, given that she lives on campus. their diet. 5Ð156. The probability that a customer will buy a Source: The Book of Odds by Michael D. Shook and Robert C. television set and buy an extended warranty is 0.03. If the Shook (New York: Penguin Putnam, Inc., 1991), p. 33. probability that a customer will purchase a television set is 0.11, find the probability that the customer will also 5Ð148. Twenty-five percent of the engineering graduates purchase the extended warranty. of a university received a starting salary of $25,000 or more. If three of the graduates are selected at random, 5Ð157. Of the members of the Blue River Health Club, find the probability that all had a starting salary of $25,000 43% have a lifetime membership and exercise regularly or more. (three or more times a week). If 75% of the club members exercise regularly, find the probability that a randomly 5Ð149. Three cards are drawn from an ordinary deck selected member is a life member, given that he or she without replacement. Find the probability of getting exercises regularly. a. All black cards b. All spades 5Ð158. The probability that it snows and the bus arrives c. All queens late is 0.023. John hears the weather forecast, and there is a 40% chance of snow tomorrow. Find the probability that 5Ð150. A coin is tossed and a card is drawn from a deck. the bus will be late, given that it snows. Find the probability of getting a. A head and a 6 5Ð159. A number of students were grouped according to b. A tail and a red card their reading ability and education. The table shows the c. A head and a club results. 5Ð151. A box of candy contains six chocolate-covered Reading ability cherries, three peppermint patties, two caramels, and two Education Low Average High strawberry creams. If a piece of candy is selected, find the probability of getting a caramel or a peppermint patty. Graduated high school 6 18 43 5Ð152. A manufacturing company has three factories: X, Y, Did not graduate 27 16 7 and Z. The daily output of each is shown below. If a student is selected at random, find these probabilities. Product Factory X Factory Y Factory Z a. The student has a low reading ability, given that the student is a high school graduate. TVs 18 32 15 b. The student has a high reading ability, given that the Stereos 6 20 13 student did not graduate. If one item is selected at random, find these probabilities. 5Ð160. At a large factory, the employees were surveyed a. It was manufactured at factory X or is a stereo. and classified according to their level of education and b. It was manufactured at factory Y or factory Z. whether or not they smoked. The data are shown in the c. It is a TV or was manufactured at factory Z. table. 5Ð153. A vaccine has a 90% probability of being effective Educational level in preventing a certain disease. The probability of getting Not high High the disease if a person is not vaccinated is 50%. In a certain Smoking school school College geographic region, 25% of the people get vaccinated. If a habit graduate graduate graduate person is selected at random, find the probability that he or she will contract the disease. Smoke 6 14 19 Do not smoke 18 7 25 5Ð154. A manufacturer makes three models of a television set, models A, B, and C. A store sells 40% of model A sets, If an employee is selected at random, find these 40% of model B sets, and 20% of model C sets. Of model probabilities. A sets, 3% have stereo sound; of model B sets, 7% have a. The employee smokes, given that he or she graduated stereo sound; and of model C sets, 9% have stereo sound. from college. If a set is sold at random, find the probability that it has b. Given that the employee did not graduate from high stereo sound. school, he or she is a smoker. Section 5Ð6 Summary 205

5Ð161. A survey done for Prevention magazine found that 5Ð164. A person has six bonds, three stocks, and two 77% of bike riders sometimes ride without a helmet. If four mutual funds. If three investments are selected, find the bike riders are randomly selected, find the probability that probability that one of each type is selected. at least one of the riders does not wear a helmet all the 5Ð165. A newspaper advertises five different movies, three time. plays, and two baseball games for the weekend. If a couple Source: Snapshot, USA Today, May 26, 1995. selects three activities, find the probability that they attend two plays and one movie. 5Ð162. A coin is tossed five times. Find the probability of getting at least one tail. 5Ð166. In an office there are three secretaries, four accountants, and two receptionists. If a committee of three 5Ð163. The U.S. Department of Health and Human is to be formed, find the probability that one of each will be Services reports that 15% of Americans have chronic selected. sinusitis. If five people are selected at random, find the probability that at least one has chronic sinusitis. Source: 100% American by Daniel Evans Weiss (New York: Poseidon Press, 1988).

Would You Bet Your Life? Revisited Today In his book Probabilities in Everyday Life, John D. McGervey states that the chance of being killed on any given commercial airline flight is almost 1 in 1 million and that the chance of being killed during a transcontinental auto trip is about 1 in 8000. The corre- sponding probabilities are 1/1,000,0000 0.000001 as compared to 1/8000 0.000125. Since the second number is 125 times greater than the first number, you have a much higher risk driving than flying across the United States.

Chapter Quiz

Determine whether each statement is true or false. If the 8. When a meteorologist says that there is a 30% chance statement is false, explain why. of showers, what type of probability is the person using? 1. Subjective probability has little use in the real world. a. Classical 2. Classical probability uses a frequency distribution to b. Empirical compute probabilities. c. Relative 3. In classical probability, all outcomes in the sample d. Subjective space are equally likely. 9. The sample space for tossing three coins consists of 4. When two events are not mutually exclusive, P(A or B) how many outcomes? P(A) P(B). a. 2 b. 4 5. If two events are dependent, they must have the same c. 6 probability of occurring. d. 8 6. An event and its complement can occur at the same time. 10. The complement of guessing five correct answers on a five-question trueÐfalse exam is Select the best answer. a. Guessing five incorrect answers b. Guessing at least one incorrect answer 7. The probability that an event happens is 0.42. What is c. Guessing at least one correct answer the probability that the event won’t happen? d. Guessing no incorrect answers a. 0.42 b. 0.58 11. When two dice are rolled, the sample space consists of c. 0 how many events? d. 1 a. 6 206 Chapter 5 Probability

b. 12 graduates are selected at random, find the probability c. 36 that all had a starting salary of $30,000 or more. d. 54 23. Five cards are drawn from an ordinary deck without Complete the following statements with the best answer. replacement. Find the probability of getting a. All red cards 12. The set of all possible outcomes of a probability b. All diamonds experiment is called the . c. All aces 13. The probability of an event can be any number 24. The probability that Sam will be accepted by the between and including and . college of his choice and obtain a scholarship is 14. If an event cannot occur, its probability is . 0.35. If the probability that he is accepted by the college is 0.65, find the probability that he will 15. The sum of the probabilities of the events in the sample obtain a scholarship given that he is accepted by space is . the college. 16. When two events cannot occur at the same time, they 25. The probability that a customer will buy a car and an are said to be . extended warranty is 0.16. If the probability that a 17. When a card is drawn, what is the probability of customer will purchase a car is 0.30, find the getting probability that the customer will also purchase the a. A jack extended warranty. b. A4 26. Of the members of the Spring Lake Bowling Lanes, c. A card less than 6 (an ace is considered above 6) 57% have a lifetime membership and bowl regularly 18. When a card is drawn from a deck, find the probability (three or more times a week). If 70% of the club of getting members bowl regularly, find the probability that a a. A diamond randomly selected member is a lifetime member given b. A 5 or a heart that he or she bowls regularly. c. A 5 and a heart 27. The probability that John has to work overtime and it d. A king rains is 0.028. John hears the weather forecast, and e. A red card there is a 50% chance of rain. Find the probability that 19. At a men’s clothing store, 12 men purchased blue golf he will have to work overtime given that it rains. sweaters, 8 purchased green sweaters, 4 purchased 28. At a large factory, the employees were surveyed and gray sweaters, and 7 bought black sweaters. If a classified according to their level of education and customer is selected at random, find the probability whether or not they attend a sports event at least once a that he purchased month. The data are shown in the table. a. A blue sweater b. A green or gray sweater Educational level c. A green or black or blue sweater High Two-year Four-year d. A sweater that was not black school college college 20. When two dice are rolled, find the probability of Sports event graduate degree degree getting a. A sum of 6 or 7 Attend 16 20 24 b. A sum greater than 8 Do not attend 12 19 25 c. A sum less than 3 or greater than 8 If an employee is selected at random, find the d. A sum that is divisible by 3 probability that e. A sum of 16 a. The employee attends sports events regularly given f. A sum less than 11 that he or she graduated from college (two- or four- 21. The probability that a person owns a microwave oven year degree). is 0.75, that a person owns a compact disk player is b. Given that the employee is a high school graduate, 0.25, and that a person owns both a microwave and a he or she does not attend sports events regularly. CD player, 0.16. Find the probability that a person 29. In a certain high-risk group, the chances of a person owns either a microwave or a CD player, but not both. having suffered a heart attack are 55%. If six people 22. Of the physics graduates of a university, 30% received are chosen, find the probability that at least one will a starting salary of $30,000 or more. If five of the have had a heart attack. Section 5Ð6 Summary 207

30. A single die is rolled four times. Find the probability of 33. A cruise director schedules four different movies, two getting at least one 5. bridge games, and three tennis games for a two-day period. If a couple selects three activities, find the 31. If 85% of all people have brown eyes and six people probability that they attend two movies and one tennis are selected at random, find the probability that at least game. one of them has brown eyes. 34. At a sorority meeting, there are six seniors, four 32. On a lunch counter, there are three oranges, five juniors, and two sophomores. If a committee of three is apples, and two bananas. If three pieces of fruit are to be formed, find the probability that one of each will selected, find the probability that one orange, one be selected. apple, and one banana are selected.

Critical Thinking Challenges

1. Consider the following problem: A con man has three For example, suppose there were three people in the coins. One coin has been specially made and has a head room. The probability that each had a different birthday on each side. A second coin has been specially made would be and on each side it has a tail. Finally, a third coin has a 365 364 363 P • • 365 3 0.992 head and a tail on it. All coins are of the same 365 365 365 3653 denomination. The con man places the three coins in his Hence, the probability that at least two of the three pocket, selects one, and shows you one side. It is heads. people will have the same birthday will be He is willing to bet you even money that it is the two- headed coin. His reasoning is that it can’t be the two- 1 0.992 0.008 tailed coin since a head is showing; therefore, there is a Hence, for k people, the formula is 50-50 chance of it being the two-headed coin. Would you take the bet? (Hint: See Exercise 1 in Data P(at least 2 people have the same birthday) P Projects.) 1 365 k 365k 2. Chevalier de Méré won money when he bet unsuspecting patrons that in four rolls of a die, he could Using your calculator, complete the following table and get at least one 6, but he lost money when he bet that in verify that for at least a 50% chance of two people 24 rolls of two dice, he could get at least a double 6. having the same birthday, 23 or more people will be Using the probability rules, find the probability of each needed. event and explain why he won the majority of the time Probability on the first game but lost the majority of the time when that at least playing the second game. (Hint: Find the probabilities Number of two have the of losing each game and subtract from one.) people same birthday 3. How many people do you think need to be in a room so 1 0.000 that two people will have the same birthday (month and 2 0.003 day)? You might think it is 366. This would, of course, guarantee it (excluding leap year), but how many people 5 0.027 would need to be in a room so that there would be a 10 90% probability that two people would be born on the 15 same day? What about a 50% probability? 20 Actually, the number is much smaller than you may 21 think. For example, if you have 50 people in a room, the 22 probability that two people will have the same birthday is 97%. If you have 23 people in a room, there is a 50% 23 probability that two people were born on the same day! 4. We know that if the probability of an event happening is The problem can be solved by using the probability 100%, then the event is a certainty. Can it be concluded rules. It must be assumed that all birthdays are equally that if there is a 50% chance of contracting a likely, but this assumption will have little effect on the communicable disease through contact with an infected answers. The way to find the answer is by using the person, there would be a 100% chance of contracting complementary event rule as P (two people having the the disease if two contacts were made with the infected same birthday) 1 P (all have different birthdays). person? Explain your answer. 208 Chapter 5 Probability

WWW Data Projects

1. Make a set of three cards—one with a red star on both Source: George W. Bright, John G. Harvey, and Margariete sides, one with a black star on both sides, and one with a Montague Wheeler, “Fair Games, Unfair Games.” Chapter 8, black star on one side and a red star on the other side. Teaching Statistics and Probability. NCTM 1981 Yearbook. With a partner, play the game described in the first Reston, Virginia: The National Council of Teachers of critical thinking challenge on page 207 one hundred Mathematics, Inc., 1981, p. 49. Used with permission. times and record the results of how many times you win 4. Often when playing gambling games or collecting items and how many times your partner wins. (Note: Do not in cereal boxes, one wonders how long will it be before change options during the 100 trials.) one achieves a success. For example, suppose there are a. Do you think the game is fair (i.e., does one person six different types of toys with one toy packaged at win approximately 50% of the time)? random in a cereal box. If a person wanted a certain toy, b. If you think the game is unfair, explain what the about how many boxes would that person have to buy probabilities might be and why. on average before obtaining that particular toy? Of 2. Take a coin and tape a small weight (e.g., part of a paper course, there is a possibility that the particular toy clip) to one side. Flip the coin 100 times and record the would be in the first box opened or that the person results. Do you think you have changed the probabilities might never obtain the particular toy. These are the of the results of flipping the coin? Explain. extremes. 3. This game is called “Diet Fractions.” Roll two dice and a. To find out, simulate the experiment using dice. Start use the numbers to make a fraction less than or equal to rolling dice until a particular number, say 3, is one. Player A wins if the fraction cannot be reduced; obtained and keep track of how many rolls are otherwise, player B wins. necessary. Repeat 100 times. Then find the average. a. Play the game 100 times and record the results. b. You may decide to use another number, such as 10 b. Decide if the game is fair or not. Explain why or different items. In this case, use 10 playing cards why not. (ace through 10 of diamonds), select a particular c. Using the sample space for two dice, compute the card (say an ace), shuffle the deck each time, deal probabilities of player A winning and player B the cards, and count how many cards are turned over winning. Do these agree with the results obtained in before the ace is obtained. Repeat 100 times, then part a? find the average. c. Summarize the findings for both experiments.