Math 3201 Chapter 3 Review

Name:__________

Multiple Choice Identify the choice that best completes the statement or answers the question.

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1. Which expression correctly describes the experimental probability P(B), where n(B) is the number of times event B occurred and n(T) is the total number of trials, T, in the experiment? A.

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B.

5P5

D. 3 : 11

B. 1 : 3

C. 3 : 4

D. 3 : 1

B. 4 : 1

C. 3 : 11

D. 3 : 4

B. 4 : 1

C. 1 : 5

D. 1 : 4

B.

5P3

C.

5 P4

D.

5P1

B. 3

C. 5

D. 7

B. 0.08%

C. 0.15%

D. 0.23%

11. Nine boys and twelve girls have signed up for a trip. Only six students will be selected to go on the trip. Determine the probability that there will be equal numbers of boys and girls on the trip. A. 17.23%

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C. 7 : 11

10. Nine boys and twelve girls have signed up for a trip. Only six students will be selected to go on the trip. Determine the probability that only boys will be on the trip. A. 0.02%

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B. 4 : 11

9. Yvonne tosses three coins. She is calculating the probability that at least one coin will land as heads. Determine the number of options where at least one coin lands as heads. A. 1

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D. P(A) = 3

8. A credit card company randomly generates temporary three-digit pass codes for cardholders. The pass code will consist of three different even digits. Determine the total number of pass codes using three different even digits. A.

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C. P(C) = 1

7. The weather forecaster says that there is an 80% probability of rain tomorrow. Determine the odds against rain. A. 4 : 5

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B. P(B) =

6. Zahra likes to go rock climbing with her friends. In the past, Zahra has climbed to the top of the wall 7 times in 28 attempts. Determine the odds against Zahra climbing to the top. A. 3 : 1

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D. P(D) =

5. Raymond has 12 coins in his pocket, and 9 of these coins are quarters. He reaches into his pocket and pulls out a coin at random. Determine the odds against the coin being a quarter. A. 1 : 4

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C. P(C) = 0.3

4. The odds in favour of Macy passing her driver’s test on the first try are 7 : 4. Determine the odds against Macy passing her driver’s test on the first try. A. 4 : 7

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B. P(B) =

3. Three events, A, B, and C, are all equally likely. If there are no other possible events, which of the following statements is true? A. P(A) = 0

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D.

2. Given the following probabilities, which event is most likely to occur? A. P(A) = 0.2

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C.

B. 22.61%

C. 27.35%

D. 34.06%

12. Nine boys and twelve girls have signed up for a trip. Only six students will be selected to go on the trip. Determine the number of ways in which there can be more girls than boys on the trip. A. 17 456

B. 25 872

C. 29 778

D. 35 910

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13. Select the events that are mutually exclusive. A. Drawing a 7 or drawing a heart from a standard deck of 52 playing cards. B. Rolling a sum of 4 or rolling an even number with a pair of four-sided dice, numbered 1 to 4. C. Drawing a black card or drawing a Queen from a standard deck of 52 playing cards. D. Rolling a sum of 8 or a sum of 11 with a pair of six-sided dice, numbered 1 to 6.

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14. Hilary draws a card from a well-shuffled standard deck of 52 playing cards. Then she draws another card from the deck without replacing the first card. Determine the probability that both cards are hearts. A.

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B.

D.

15. Min draws a card from a well-shuffled standard deck of 52 playing cards. Then she puts the card back in the deck, shuffles again, and draws another card from the deck. Determine the probability that both cards are face cards. A.

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C.

B.

C.

D.

16. Select the events that are dependent. A. Drawing a face card from a standard deck of 52 playing cards, putting it back, and then drawing another face card. B. Rolling a 4 and rolling a 3 with a pair of six-sided dice, numbered 1 to 6. C. Drawing a heart from a standard deck of 52 playing cards, putting it back, and then drawing another heart. D. Rolling a 3 and having a sum greater than 5 with a pair of six-sided dice, numbered 1 to 6.

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17. There are 60 males and 90 females in a graduating class. Of these students, 30 males and 50 females plan to attend a certain university next year. Determine the probability that a randomly selected student plans to attend the university. A. 0.41

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D. 0.59

B. 19.64%

C. 26.47%

D. 32.13%

19. A three-colour spinner is spun, and a die is rolled. Determine the probability of spinning blue and rolling a 4. A. 1.24%

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C. 0.53

18. Paul has four loonies, three toonies, and five quarters in his pocket. He needs two quarters for a parking meter. He reaches into his pocket and pulls out two coins at random. Determine the probability that both coins are quarters. A. 15.15%

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B. 0.47

B. 5.56%

C. 7.17%

D. 9.82%

20. Select the independent events. A. B. C. D.

P(A) = 0.67, P(B) = 0.12, and P(A  B) = 0.086 P(A) = 0.83, P(B) = 0.4, and P(A  B) = 0.378 P(A) = 0.4, P(B) = 0.91, and P(A  B) = 0.364 P(A) = 0.2, P(B) = 0.32, and P(A  B) = 0.046

Problem

1. A hockey game has ended in a tie after a 5 min overtime period, so the winner will be decided by a shootout. The coach must decide whether Jules or Vicki should go first in the shootout. The coach would prefer to use her best scorer first, so she will base her decision on the players’ shootout records. Player Attempts Goals Scored Jules 15 7 Vicki 19 12 Who should go first? Show your work.

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2. Three people are running for president of the student council. The polls show Denis has a 55% chance of winning, Cyndi has a 25% chance of winning, and Chris has a 20% chance of winning. a) What are the odds in favour of each person winning? Show your work. b) Suppose that Chris withdraws and offers his support to Cyndi. Further suppose that his supporters also switch to Cyndi. What are the odds in favour of Cyndi winning now? 3. Atian, Sam, Phuong, Mike, and Tariq are competing with ten other boys to be on their school’s cross-country team. All the boys have an equal chance of winning the trial race. Determine the probability that Atian, Sam, Phuong, Mike, and Tariq will place first, second, third, fourth, and fifth, in any order. Show your work. 4. Conner hosts a morning radio show in Kenora. To advertise his show, he is holding a contest at a local mall. He spells out ONTARIO with letter tiles. Then he turns the tiles face down and mixes them up. He asks Gabe to arrange the tiles in a row and turn them face up. If the row of tiles spells ONTARIO, Gabe will win a new car. Determine the probability that Gabe will win the car. Show your work. 5. There are 11 players on a baseball team, all with roughly equal athletic ability. The coach has decided to choose the players who will play the four infield positions (first base, second base, third base, and shortstop) randomly. Tori and Brittany are on the team. Determine the odds in favour of Tori and Brittany being chosen to play in the infield. Show your work. 6. A student council consists of 12 girls and 8 boys. To form a subcommittee, 4 students are randomly selected from the council. Determine the odds in favour of 3 girls and 1 boy being on the subcommittee. Show your work. 7. A car manufacturer keeps a database of all the cars that are available for sale at all the dealerships in Western Canada. For model A, the database reports that 36% have heated leather seats, 41% have a sunroof, and 52% have neither. Determine the probability of a model A car at a dealership having both heated leather seats and a sunroof. Show your work. 8. A survey reported that 42% of households have one or more dogs, 28% have one or more cats, and 39% have neither dogs nor cats. Suppose that a household is selected at random. Determine the probability that there are cats but no dogs in the household. Show your work. 9. On Thursday, the weather forecaster says that there is a 40% chance of rain on Friday and a 70% chance of rain on Saturday. The forecaster also says that there is a 20% chance of rain on both Friday and Saturday. Determine the probability that there will be rain on Friday or on Saturday. Show your work. 10. Rowan is the coach of a junior ultimate team. Based on the team’s record, it has a 80% chance of winning on calm days and a 60% chance of winning on windy days. Tomorrow, there is a 60% chance of high winds. There are no ties in ultimate. What is the probability that Rowan’s team will win tomorrow? Show your work. 11. Each day, Julia’s math teacher gives the class a warm-up question. It is a true-false question 20% of the time and a multiple-choice question 80% of the time. Julia gets 70% of the true-false questions correct, and 90% of the multiple-choice questions correct. Julia answers today’s question correctly. What is the probability that it was a multiple-choice question? Show your work. 12. Mena remembers to set her alarm clock 71% of the time. When she does remember to set her alarm clock, the probability that she will be late for school is 0.10. When she does not remember to set it, the probability that she will be late for school is 0.80. Mena was late today. What is the probability that she remembered to set her alarm clock? Show your work. 13. The probability that a plane will leave Winnipeg on time is 0.80. The probability that a plane will leave Winnipeg on time and arrive in Calgary on time is 0.42. Determine the probability that a plane will arrive in Calgary on time, given that it left Winnipeg on time. Show your work.

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14. Lulu goes to the gym five days a week. Each day, she does a cardio workout using either a stepper or an elliptical walker. She follows this with a strength workout using either free weights or the weight machines. Lulu randomly chooses which cardio workout and which strength workout to do each day. Determine the probability that Lulu will use a stepper and the weight machines the next day. Show your work. 15. Elin estimates that her probability of passing French is 0.6 and her probability of passing chemistry is 0.8. Determine the probability that Elin will pass French but fail chemistry. Show your work.

Answer Section MULTIPLE CHOICE 1.B 2.D 3.B 4.A 5.B 6.A 7.D 8.B 9.D 10.C 11.D 12.B 13.D 14.B 15.B 16.D 17.A 18.A 19.B 20.C PROBLEM 1. Jules has 15 attempts and has scored 7 goals. This means that she has 15 – 7 or 8 attempts where she did not score. The odds in favour of her scoring are 7 : 8. Vicki has 19 attempts and has scored 12 goals. This means that she has 19 – 12 or 7 attempts where she did not score. The odds in favour of her scoring are 12 : 7. The probability that Jules will score is

or about 0.467.

The probability that Vicki will score is

or about 0.632.

Since 0.632 > 0.467, there is a better chance that Vicki will score. Therefore, Vicki should go first. 2. a) The odds in favour of Denis winning are 55 : (100 – 55). This is equal to 55: 45 or 11 : 9. The odds in favour of Cyndi winning are 25 : (100 – 25). This is equal to 25 : 75 or 1 : 3. The odds in favour of Chris winning are 20 : (100 – 20). This is equal to 20 : 80 or 1 : 4. b) If Chris’ 20% support goes to Cyndi, then her support will now be 45%, and the odds in favour of Cyndi winning will be the same as the odds against Denis winning. So, the odds in favour of Cyndi winning are 45 : 55 or 9 : 11. 3. Atian, Sam, Phuong, Mike, and Tariq can place first, second, third, fourth, or fifth, in any order. There are 5! or 120 ways in which five runners can place in five positions. There are 15P5 ways that 15 runners can place first, second, third, fourth, or fifth. There are 360 360 possible outcomes. P(A, S, P, M, and T place 1, 2, 3, 4, or 5) = P(A, S, P, M, and T place 1, 2, 3, 4, or 5) =

or

The probability that Atian, Sam, Phuong, Mike, and Tariq will place in the top five positions is 4. There are 7 letters in total: 2 O’s and 5 other letters. Let L represent the total number of ways to arrange the letters.

or about 0.03%.

You can spell ONTARIO in just 1 way. So, there is only 1 favourable outcome. P(winning the car) = P(winning the car) =

This is the total number of outcomes. Let R represent the number of ways to spell ONTARIO. R=1

The probability that Gabe will win the car is .

5 5. Let T represent Tori and Brittany being chosen to play in the infield. Let O represent all possible infield lineups. The number of ways to arrange Tori and Brittany in the infield positions is 4P2. The number of ways to arrange the other 9 players in the remaining 2 infield positions is 9P2. Therefore, the total number of infield lineups that include Tori and Brittany is (2  3!) 9P2.

The total number of infield lineups possible is 11P4

The probability can now be determined: The probability that Tori and Brittany will both play in the infield is

.

Therefore the odds in favour of this event are 6 : (55 – 6) or 6 : 49.

6. Let T represent three girls and one boy being chosen to form a subcommittee, and let S represent all possible subcommittees. In this example, order is not important. The number of ways to arrange three girls and one boy from 12 girls and 8 boys is 12C3  8C1.

The number of ways to arrange 20 people in a four-person committee is 20C4.

The probability can now be determined:

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7. Let A represent the universal set of all model A cars. Let L represent model A cars with heated leather seats. Let S represent model A cars with a sunroof. P(L  S) = 100% – 52% P(L  S) = 48%

The probability of a model A car at a dealership having both heated seats and a sunroof is 29%. 8. Let D represent the households that have one or more dogs, and let C represent the households that have one or more cats. P(D) = 42% P(C) = 28% P(D  C) = 100% – 39% P(D  C) = 61%

P(C \ D) = P(C) – P(D  C) P(C \ D) = 28% – 9% P(C \ D) = 19% The probability that a household has one or more cats, but no dogs, is 19%. 9. Let F represent rain on Friday and let S represent rain on Saturday. P(F) = 40% P(S) = 70% P(F  S) = 20% P(F  S) = P(F) + P(S) – P(F  S) P(F  S) = 40% + 70% – 20% P(F  S) = 90% The probability that it will rain on Friday or on Saturday is 90%. 10. P(windy) is 60%, so P(calm) is 100% – 60% or 40%. P(win | windy) = 60% P(lose | windy) = 100% – 60% or 40% P(win | calm) = 80% P(lose | calm) = 100% – 80% or 20%

P(win) = P(windy  win) + P(calm  win) P(win) = 0.36 + 0.32 P(win) = 0.68 The probability that Rowan’s team will win tomorrow is 68%.

7 11. Let T represent a true or false question, and let M represent a multiple-choice question. Let C represent a correct question. P(T  C) = P(T) P(C | T) P(T  C) = 0.2 0.7 P(T  C) = 0.14 P(M  C) = P(M) P(C | M) P(M  C) = 0.8 0.9 P(M  C) = 0.72 P(C) = 0.14 + 0.72 P(C) = 0.86

The probability the question was multiple-choice is

, or about 0.837 or 83.7%.

12. Let S represent Mena remembering to set her alarm, and let N represent Mena not remembering to set her alarm. Let L represent Mena being late for school. P(S  L) = P(S) P(L | S) P(S  L) = 0.71 0.10 P(S  L) = 0.071 P(N  L) = P(N) P(L | N) P(N  L) = 0.29 0.80 P(N  L) = 0.232 P(C) = 0.071 + 0.232 P(C) = 0.303

The probability Mena’s alarm clock was set is

, or about 0.234 or 23.4%.

13. Let L represent a plane leaving from Winnipeg on time, and let A represent a plane arriving in Calgary on time. P(L) = 0.80 P(L  A) = 0.42

The probability that a plane will arrive in Calgary on time, given it left Winnipeg on time, is 0.525 or 52.5%. 14. Let S represent Lulu using a stepper. Let M represent Lulu using weight machines. P(S) = P(M) = P(S  M) = P(S)

P(M)

P(S  M) = P(S  M) = The probability Lulu will use a stepper and the weight machines for her next workout is

, or 0.25 or 25%.

8 15. Let F represent passing French, and let C represent passing chemistry. P(F) = 0.6 P(C) = 0.2 P(F  C) = P(F) P(C) P(F  C) = 0.6 0.2 P(F  C) = 0.12 The probability that Elin will pass French but fail Chemistry is 0.12, or 12%.