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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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