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Ch6 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. A coin is tossed three times. What is the probability of tossing at least two heads? a. c. b. d. ____ 2. Two standard dice are rolled. What is the probability of rolling a pair (both the same number)? a. c. b. d. ____ 3. Which of the following statements is false? a. “Theoretical probability” is also called “classical probability” or “a priori probability.” b. “Experimental probability” is another term for “relative-frequency probability.” c. A “sample space” is the set of all possible outcomes of a probability experiment. d. “Empirical probability” is another term for “subjective probability.” ____ 4. Two identical spinners each have five equal sectors that are numbered 1 to 5. What is the probability of a total less than 9 when you spin both these spinners? a. c. b. d. ____ 5. If the odds in favour of snow tomorrow are 4:7, what is the probability of snow tomorrow? a. b. c. d. ____ 6. What is the probability of randomly selecting a blue sock from an unorganized sock drawer if the odds in favour of not picking a blue sock are 3 to 7? a. b. c. d. ____ 7. A group of volleyball players consists of four grade-11 students and six grade-12 students. If six players are chosen at random to start a match, what is the probability that three will be from each grade? a. b. c. d. ____ 8. Without looking, Jenny randomly selects two socks from a drawer containing four blue, three white, and five black socks, none of which are paired up. What is the probability that she chooses two socks of the same colour? a. b. c. d. ____ 9. If two standard dice are rolled, what is the probability of doubles (both dice showing the same number) or a total of 7? a. b. c. d. ____ 10. In a particular class, the probability of a student having blue eyes is 0.3, of having both blue eyes and blonde hair is 0.2, and of having neither blue eyes nor blonde hair is 0.5. What is the probability that a student in this class has blonde hair? a. 0.5 b. 0.2 c. 0.3 d. 0.1 Short Answer 11. Explain the differences between the empirical probability, theoretical probability, and subjective probability. 12. Explain the meaning of the terms in the probability formula, . 13. Two coins are tossed simultaneously. What is the probability of tossing a head and a tail? 14. A coin is tossed 3 times. What is the probability of tossing at least one head? 15. Two standard dice are rolled. What is the probability that a sum less than 7 is not rolled? 16. What are the odds in favour of July 1st being a Tuesday? 17. What are the odds in favour of a total greater than 9 in a given roll of two standard dice? 18. If the odds are 9:1 against the next car you see being red, what proportion of cars in your area are red? 19. A club with eight members from grade 11 and five members from grade 12 is to elect a president, vice-president, and secretary. What is the probability (as a percentage to one decimal place) that grade 12 students will be elected for all three positions, assuming that all club members have an equal chance of being elected? 20. If a CD player is programmed to play the CD tracks in random order, what is the probability that it will play six songs from a CD in order from your favourite to your least favourite? 21. What is the probability that at least two people in a class of 30 students have the same birthday? Assume that no one in the class was born on February 29. 22. If a satellite launch has a 97% chance of success, what is the probability of three consecutive successful launches? 23. Carrie is a kicker on her rugby team. She estimates that her chances of scoring on a penalty kick during a game are 75% when there is no wind, but only 60% on a windy day. If the weather forecast gives a 55% probability of windy weather today, what is the probability of Carrie scoring on a penalty kick in a match this afternoon? 24. A bag contains three white marbles, five green marbles, and two red marbles. What are the odds in favour of randomly picking both red marbles in the first two tries? Assume that the first marble picked is not put back into the bag. 25. If the probability of the Rangers defeating the Eagles in a hockey game is , what is the probability that the Rangers will win two consecutive games against the Eagles? 26. Statsville has two computer-controlled traffic lights on the road between the main street and the highway. The probability of getting a red light at the first traffic light is 0.45, and the probability of getting a red light at the second one is 0.20 if you had been stopped by a red light at the first one. What is the probability of being stopped by red lights at both intersections? 27. The probability that Jacqueline will be elected to the students’ council is 0.6, and the probability that she will be selected to represent her school in a public-speaking contest is 0.75. The probability of Jacqueline achieving both of these goals is 0.5. a) Are these two goals mutually exclusive? Explain your answer. b) What is the probability that Jacqueline is either elected to the students’ council or picked for the public-speaking contest? c) What is the probability that she fails to be selected for either the students’ council or the public-speaking contest? 28. The probability that Sarjay will play golf today is 60%, the probability that he will play golf tomorrow is 75%, and the probability that he will play golf on both days is 50%. What is the probability that he does not play golf on either day? 29. In Statsville, the probability of a teenager listening to the classic-rock radio station is 42%, while the probability of a teenager listening to the new-country radio station is 36%. If 13% of the teenagers surveyed listen to both these stations, what is the probability that a given teenager listens to neither of the two stations? 30. If 28% of the population of Statsville wears contact lenses, 37% have blue eyes, and 9% are blue-eyed people who wear contact lenses, what is the probability that a randomly selected resident has neither blue eyes nor contact lenses? Problem 31. Use a tree diagram to explain why the probability that a family with four children has either all girls or all boys is , assuming that the probability of having a boy equals the probability of having a girl. 32. A stable has 15 horses available for trail rides. Of these horses, 6 are all brown, 5 are mainly white, and the rest are black. If Jasmine selects one at random, what is the probability that this horse will a) be black? b) not be black? c) be either black or brown? 33. A number is chosen randomly from the first 20 natural numbers (1 to 20). If event A = {a multiple of 5}, what is the value of P(A)? 34. Describe how to calculate the odds against an event happening when you know the probability of the event occurring. Use a numerical example to illustrate your explanation. 35. Explain how you would calculate the tomorrow’s “probability of precipitation” if the odds against precipitation tomorrow are 4:1. 36. Explain why odds of 4 to 5 in favour of an event occurring have a different meaning than the same event having a probability of . 37. If you were to toss four coins, what are the odds in favour of at least two landing heads up? 38. Suppose you randomly draw two marbles, without replacement, from a bag containing six green, four red, and three black marbles. a) Draw a tree diagram to illustrate all possible outcomes of this draw. b) Determine the probability that both marbles are red. c) Determine the probability that you pick at least one green marble. 39. Len just wrote a multiple-choice test with 15 questions, each having four choices. Len is sure that he got exactly 9 of the first 12 questions correct, but he guessed randomly on the last 3 questions. What is the probability that he will get at least 80% on the test? 40. Leela has five white and six grey huskies in her kennel. If a wilderness expedition chooses a team of six sled dogs at random from Leela’s kennel, what is the probability the team will consist of a) all white huskies? b) all grey huskies? c) three of each colour? 41. Six friends go to their favourite restaurant, which has ten entrees on the menu. If the friends are equally likely to pick any of the entrees, what is the probability that at least two of them will order the same one? 42. To get out of jail free in the board game MONOPOLY®, you have to roll doubles with a pair of standard dice. Determine the odds in favour of getting out of jail on your first or second roll. 43. At an athletic event, athletes are tested for steroids using two different tests. The first test has a 93.0% probability of giving accurate results, while the second test is accurate 87.0% of the time. For a sample that does contain steroids, what is the probability that a) neither test shows that steroids are present? b) both tests show that steroids are present? c) at least one of the tests detects the steroids? 44. A test for the presence of E. coli in water detects the bacteria 97% of the time when the bacteria is present, but also gives a false positive 2% of the time, wrongly indicating the presence of E. coli in uninfected water. If 10% of the water samples tested contain E. coli, what is the probability that a test result indicating the presence of the bacteria is accurate? 45. A study on the effects that listening to loud music through headphones had on teenagers’ hearing found that 12% of those teenagers in the sample who did listen to music in this way showed signs of hearing problems. If 60% of the sample reported that they listened to loud music on headphones regularly, and 85% of the sample were found not to have hearing problems, are the events {having hearing problems} and {listening to loud music on headphones} independent? Explain your reasoning. 46. The Raptors are underdogs in their series with the Celtics: the odds in favour of the Raptors winning each game are 3:5. If the Raptors lose the first game of a best-of-five series, what is the probability that they will win the series? 47. Explain the difference between mutually exclusive events and independent events using an example of each to illustrate your answer. In your explanation, show why probabilities are added for a mutually exclusive set of events and are multiplied for independent events. 48. Of the students at Statsville High School who take both mathematics and English, 83% pass English, 65% pass both mathematics and English, and 5% fail both subjects. What is the probability of passing mathematics and failing English for this group of students? Explain your reasoning. 49. If a survey on teenage readers of popular magazines shows that 38% subscribe to Teen People, 47% subscribe to Cool Life, and 35% subscribe to neither magazine, what is the probability that a randomly selected teenager a) subscribes to both magazines? b) subscribes to either one magazine or both? c) subscribes to only one of the two magazines? 50. A survey of 50 female high-school athletes collected the following data. Team Number of Athletes Field hockey 23 Volleyball 16 Rugby 29 Both rugby and field hockey 8 Both rugby and volleyball 9 Both field hockey and volleyball 7 All three teams 6 a) Draw a Venn diagram to illustrate the above data. b) Determine the probability that a randomly selected athlete from this sample will play either rugby or field hockey. c) What is the probability that a randomly selected athlete will play on only one of the three sports teams? d) Determine the probability that a randomly selected rugby player also plays volleyball. e) Determine the probability that a randomly selected athlete who does not play rugby is on the field-hockey team. Ch6 Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Calculating probability 2. ANS: A PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Calculating probability 3. ANS: D PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Probability concepts 4. ANS: C PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Calculating probability 5. ANS: C PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.2 TOP: Odds and probability 6. ANS: D PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.2 TOP: Odds and probability 7. ANS: C PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.3 TOP: Calculating probability 8. ANS: B PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.3 TOP: Calculating probability 9. ANS: B PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.5 TOP: Mutually exclusive events 10. ANS: B PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.5 TOP: Mutually exclusive events SHORT ANSWER 11. ANS: Empirical probability is based on experimental results and direct observations of events. Theoretical probability is based on mathematical calculations and the analysis of the possible outcomes. Subjective probability is based on past experience or informed guesswork. PTS: 1 REF: Communication OBJ: Section 6.1 TOP: Probability concepts 12. ANS: P represents probability. A represents the event A. The letter n stands for “number” and n(A) represents the number of outcomes in which event A occurs. S represents the sample space, and n(S) is the number of outcomes in the sample space (the total number of possible outcomes). PTS: 1 REF: Communication OBJ: Section 6.1 TOP: Probability concepts 13. ANS: PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Calculating probability 14. ANS: PTS: 1 REF: Knowledge & Understanding OBJ: Section 6.1 TOP: Calculating probability 15. ANS: PTS: 1 REF: Applications OBJ: Section 6.1 TOP: Calculating probability 16. ANS: 1:6 PTS: 1 REF: Applications OBJ: Section 6.2 TOP: Calculating odds 17. ANS: 1:5 PTS: 1 REF: Applications OBJ: Section 6.2 TOP: Calculating odds 18. ANS: 10% PTS: 1 REF: Applications OBJ: Section 6.2 TOP: Calculating odds 19. ANS: 3.5% PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 20. ANS: or about 0.001 39 PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 21. ANS: about 0.7063 PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 22. ANS: About 91% PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 23. ANS: PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 24. ANS: The probability of picking the two red marbles in the first two picks is . Therefore, the odds in favour of picking the two red marbles are 1:44. PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 25. ANS: PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 26. ANS: 9.0% PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 27. ANS: a) The two goals cannot be mutually exclusive since the probability of achieving both is 0.5. b) 0.85 c) 0.15 PTS: 1 REF: Applications | Communication OBJ: Section 6.5 TOP: Mutually exclusive events 28. ANS: 15% PTS: 1 REF: Applications OBJ: Section 6.5 TOP: Mutually exclusive events 29. ANS: 35% PTS: 1 REF: Applications OBJ: Section 6.5 TOP: Mutually exclusive events 30. ANS: 44% PTS: 1 REF: Applications OBJ: Section 6.5 TOP: Mutually exclusive events PROBLEM 31. ANS: Two of the sixteen branches represent outcomes with either four boys or four girls. Each of the sixteen outcomes is equally likely, so the probability of having all girls or all boys is . PTS: 1 REF: Applications | Communication OBJ: Section 6.1 TOP: Calculating probability | Tree diagrams 32. ANS: a) b) c) PTS: 1 REF: Applications OBJ: Section 6.1 TOP: Calculating probability 33. ANS: There are four multiples of 5 among the first 20 natural numbers. Therefore, n(A) = 4 and n(A) = 16. PTS: 1 REF: Applications OBJ: Section 6.1 TOP: Calculating probability 34. ANS: Answers may vary. Students should make the key point that the odds against the event happening are given by the ratio . For example, if the probability of the event occurring is , then the probability of it not happening is , and the odds against the event are , or 1:2. PTS: 1 REF: Communication OBJ: Section 6.2 TOP: Calculating odds 35. ANS: If the odds against having rain are 4:1, then the probability of not having rain is . Therefore, the probability of precipitation must be , or 20%. PTS: 1 REF: Applications | Communication OBJ: Section 6.2 TOP: Calculating odds 36. ANS: The odds of 4 to 5 mean that the event has a probability of . Thus, odds of 4 to 5 are quite different from a probability of . PTS: 1 REF: Communication OBJ: Section 6.2 TOP: Calculating odds 37. ANS: P(tossing at least two heads) 1 P(no heads) P(one head) 1 Therefore, the odds in favour of tossing at least two heads are 11:5. PTS: 1 REF: Applications OBJ: Section 6.2 TOP: Calculating odds 38. ANS: a) b) c) PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 39. ANS: A score of 80% requires getting 12 out of the 15 questions right. If Len answered 9 out of the first 12 questions correctly, he can score 80% only if he guessed all 3 of the remaining questions correctly. Therefore Len has only about a 1.6% chance of getting 80% on the test. PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 40. ANS: a) The probability is 0 since there are only 5 white huskies available. b) Since there are 11 dogs altogether, the team can be chosen in ways. However, there are only 6 grey huskies, so there is only one way of picking an all grey team. The probability of randomly selecting this team from the 11 dogs is c) PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 41. ANS: This question is similar to the birthday problem in Example 3 on p.323 of the student textbook. If none of the friends pick the same entree, there are ways to select their meals. The probability of this event is Therefore, the probability that at least two will order the same entree is 1 – 0.1512 = 0.8488, or about 84.9%. PTS: 1 REF: Applications OBJ: Section 6.3 TOP: Calculating probability 42. ANS: The probability of rolling doubles on the first roll is . The probability of not rolling doubles on the first roll is . Therefore, the probability of rolling doubles on the second roll is . The probability of rolling doubles on the first roll or the second roll is . Thus, the odds in favour of getting out of jail on either the first or second try are 11:25. PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 43. ANS: a) b) c) PTS: 1 REF: Applications | Communication OBJ: Section 6.4 TOP: Dependent and independent events 44. ANS: If 10% of the water samples contain E. coli and the test is 97% effective, then However, 90% of the samples do not contain E. coli and these samples will test positive 2% of the time, so The overall probability of a positive test result is 0.097 + 0.018 = 0.115. Therefore, the conditional probability formula gives The probability is 84% that a positive test result is accurate. PTS: 1 REF: Thinking/Inquiry/Problem Solving OBJ: Section 6.4 TOP: Dependent and independent events 45. ANS: If events A and B are independent, . Since P(hearing problems) = 0.15 and P(listening to loud music on headphones) = 0.60, then However, the observed probability of having hearing problems and listening to loud music on headphones is 0.12, which is significantly higher than 0.09. Therefore, these two events cannot be independent if the survey results are accurate. PTS: 1 REF: Communication | Thinking/Inquiry/Problem Solving OBJ: Section 6.4 TOP: Dependent and independent events 46. ANS: If the odds are 3:5, then the probability of the Raptors winning each game is . They have lost the first game, so only four games remain. Case 1: Win in four games To win the series in four games out of five the Raptors must win the next three consecutive games. There is only one way to win three in a row, so the probability of winning in four games is , or about 0.0527. Case 2: Win in five games The Raptors must lose only one of the second, third, or fourth games. Thus, there are three possible ways to win the series in five games, and the probability is , or about 0.0989. Therefore, the probability of the Raptors winning the series is 0.0527 + 0.0989 = 0.1516 or about 15.2%. PTS: 1 REF: Applications OBJ: Section 6.4 TOP: Dependent and independent events 47. ANS: Answers may vary. Students should make the key point that mutually exclusive events cannot occur at the same time, while some independent events can. The probability of an independent event is not affected by the occurrence of other events. PTS: 1 REF: Communication OBJ: Section 6.5 TOP: Mutually exclusive events 48. ANS: The 83% of students who pass English include the 65% who pass both English and mathematics. So, 85% – 65% = 18% pass English but fail mathematics. The four possibilities—pass both, pass English only, pass mathematics only, and fail both—are mutually exclusive, so the percents for these categories must add to 100. Therefore, the probability of passing mathematics and failing English for this group of students is 100% – (65% + 18% + 5%) = 12%. PTS: 1 REF: Applications | Communication OBJ: Section 6.5 TOP: Mutually exclusive events 49. ANS: a) Using a Venn diagram, b) c) PTS: 1 REF: Applications | Communication OBJ: Section 6.5 TOP: Mutually exclusive events 50. ANS: a) b) Only 6 athletes do not play either field hockey or rugby. Therefore, the probability of selecting an athlete who plays either sport is = 0.88. c) From the Venn diagram, 38 athletes play on only one of the three sports teams. Therefore, P(only one team) = , or 0.76. d) Venn Diagram Method The Venn diagram above shows that 9 volleyball players also play on the rugby team. Therefore, . Conditional Probability Method e) Venn Diagram Method The Venn diagram above shows that 15 out of the 21 athletes not on the rugby team play field hockey. Therefore, Conditional Probability Method PTS: 1 REF: Applications | Communication OBJ: Section 6.5 TOP: Mutually exclusive events

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posted: | 3/10/2012 |

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