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  • Chapter 14Probability: Homework

    SAMPLE SPACES AND PROBABILITY

    In problems 1 - 6, write a sample space for the given experiment.

    A die is rolled.

    (14.1){1,2,3,4,5,6}
    Exercise 2.

    A penny and a nickel are tossed.

    A die is rolled, and a coin is tossed.

    (14.2){1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T}
    Exercise 4.

    Three coins are tossed.

    Two dice are rolled.

    Table 14.1.
     123456
    1(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 6)
    2(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6)
    3(3, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 6)
    4(4, 1)(4, 2)(4, 3)(4, 4)(4, 5)(4, 6)
    5(5, 1)(5, 2)(5, 3)(5, 4)(5, 5)(5, 6)
    6(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)
    Exercise 6.

    A jar contains four marbles numbered 1, 2, 3, and 4. Two marbles are drawn.

    In problems 7 - 12, a card is selected from a deck. Find the following probabilities.

    (14.3)P(an ace)
    (14.4)4/52
    Exercise 8.
    (14.5)P(a red card)
    (14.6)P(a club)
    (14.7)13/52
    Exercise 10.
    (14.8)P(a face card)
    (14.9)P(a jack or spade)
    (14.10)16/52
    Exercise 12.
    (14.11)P(a jack and a spade)

    A jar contains 6 red, 7 white, and 7 blue marbles. If a marble is chosen at random, find the following probabilities.

    (14.12)P(red)
    (14.13)6/20
    Exercise 14.
    (14.14)P(white)
    (14.15)P(red or blue)
    (14.16)13/20
    Exercise 16.
    (14.17)P(red and blue)

    Consider a family of three children. Find the following probabilities.

    (14.18)P(two boys and a girl)
    (14.19)3/8
    Exercise 18.
    (14.20)P(at least one boy)
    (14.21)P(children of both sexes)
    (14.22)6/8
    Exercise 20.
    (14.23)P(at most one girl)

    Two dice are rolled. Find the following probabilities.

    (14.24)P(the sum of the dice is 5)
    (14.25)4/36
    Exercise 22.
    (14.26)P(the sum of the dice is 8)
    (14.27)P(the sum is 3 or 6)
    (14.28)7/36
    Exercise 24.
    (14.29)P(the sum is more than 10)

    A jar contains four marbles numbered 1, 2, 3, and 4. If two marbles are drawn, find the following probabilities.

    (14.30)P(the sum of the number is 5)
    (14.31)4/12
    Exercise 26.
    (14.32)P(the sum of the numbers is odd)
    (14.33)P(the sum of the numbers is 9)
    0
    Exercise 28.
    (14.34)P(one of the numbers is 3)

    MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE

    Determine whether the following pair of events are mutually exclusive.

    (14.35)A={A person earns more than $25,000}
    (14.36)B={A person earns less than $20,000}
    Yes
    Exercise 30.

    A card is drawn from a deck.

    (14.37)C={It is a King}D={It is a heart}.

    A die is rolled.

    (14.38)E={An even number shows}
    (14.39)F={A number greater than 3 shows}
    No
    Exercise 32.

    Two dice are rolled.

    (14.40)G={The sum of dice is 8}
    (14.41)H={One die shows a 6}

    Three coins are tossed.

    (14.42)I={Two heads come up}
    (14.43)J={At least one tail comes up}
    No
    Exercise 34.

    A family has three children.

    (14.44)K={First born is a boy}
    (14.45)L={The family has children of both sexes}

    Use the addition rule to find the following probabilities.

    A card is drawn from a deck, and the events C and D are as follows:

    (14.46)C={It is a king}
    (14.47)D={It is a heart}

    Find P(C  or   D).

    (14.48)16/52
    Exercise 36.

    A die is rolled, and the events E and F are as follows:

    (14.49)E={An even number shows}
    (14.50)F={A number greater than 3 shows}

    Find P(E   or   F).

    Two dice are rolled, and the events G and H are as follows:

    (14.51)G={The sum of dice is 8}
    (14.52)H={Exactly one die shows a 6}

    Find P(G  or   H).

    (14.53)13/36
    Exercise 38.

    Three coins are tossed, and the events I and J are as follows:

    (14.54)I={Two heads come up}
    (14.55)J={At least one tail comes up}

    Find P(I   or  J).

    At De Anza college, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?

    40%
    Exercise 40.

    This quarter, there is a 50% chance that Jason will pass Accounting, a 60% chance that he will pass English, and 80% chance that he will pass at least one of these two courses. What is the probability that he will pass both Accounting and English?

    The following table shows the distribution of Democratic and Republican U.S. Senators by gender.

    Table 14.2.
     MALES(M)FEMALES(F)TOTAL
    DEMOCRATS(D)39443
    REPUBLICANS(R)52557
    TOTALS919100

    Use this table to determine the following probabilities.

    (14.56)P(M   and  D)
    (14.57)39/100
    Exercise 42.
    (14.58)P(F   and   R)
    (14.59)P(M   or  D)
    (14.60)95/100
    Exercise 44.
    (14.61)P(F   or   RC)
    (14.62)P(MC   or  R)
    (14.63)61/100
    Exercise 46.
    (14.64)P(M  or   F)

    Again, use the addition rule to determine the following probabilities.

    If P(E)=.5 and P(F)=.4 and E and F are mutually exclusive, find P(E   and   F).

    0
    Exercise 48.

    If P(E)=.4 and P(F)=.2 and E and F are mutually exclusive, find P(E  or  F).

    If P(E)=.3 and P(E   or   F)=.6 and P(E   and   F)=.2, find P(F).

    0.5
    Exercise 50.

    If P(E)=.4, P(F)=.5 and P(E  or  F)=.7, find P(E   and  F).

    CALCULATING PROBABILITIES USING TREE DIAGRAMS AND COMBINATIONS

    Two apples are chosen from a basket containing five red and three yellow apples. Draw a tree diagram below, and find the following probabilities.

    (14.65)P(both red)
    (14.66)20/56
    Exercise 52.
    (14.67)P(one red, one yellow)
    (14.68)P(both yellow)
    (14.69)6/56
    Exercise 54.
    (14.70)P(First red and second yellow)

    A basket contains six red and four blue marbles. Three marbles are drawn at random. Find the following probabilities using the method shown in Example 13.2. Do not use combinations.

    (14.71)P(All three red)
    (14.72)1/6
    Exercise 56.
    (14.73)P(two red, one blue)
    (14.74)P(one red, two blue)
    (14.75)3/10
    Exercise 58.
    (14.76)P(first red, second blue, third red)

    Three marbles are drawn from a jar containing five red, four white, and three blue marbles. Find the following probabilities using combinations.

    (14.77)P(all three red)
    (14.78)10/220
    Exercise 60.
    (14.79)P(two white and 1 blue)
    (14.80)P(none white)
    (14.81)56/220
    Exercise 62.
    (14.82)P(at least one red)

    A committee of four is selected from a total of 4 freshmen, 5 sophomores, and 6 juniors. Find the probabilities for the following events.

    At least three freshmen.

    (14.83)45/1365
    Exercise 64.

    No sophomores.

    All four of the same class.

    (14.84)21/1365
    Exercise 66.

    Not all four from the same class.

    Exactly three of the same class.

    (14.85)324/1365
    Exercise 68.

    More juniors than freshmen and sophomores combined.

    Five cards are drawn from a deck. Find the probabilities for the following events.

    Two hearts, two spades, and one club.

    (14.86)79092/2,598,960
    Exercise 70.

    A flush of any suit(all cards of a single suit).

    A full house of nines and tens(3 nines and 2 tens).

    (14.87)24/2,598,960
    Exercise 72.

    Any full house.

    A pair of nines and tens.

    (14.88)1,584/2,598,960
    Exercise 74.

    Two pairs

    Do the following birthday problems.

    If there are five people in a room, what is the probability that no two have the same birthday?

    0.973
    Exercise 76.

    If there are five people in a room, what is the probability that at least two people have the same birthday?

    CONDITIONAL PROBABILITY

    Do the following problems using the conditional probability formula: _autogen-svg2png-0119.png.

    A card is drawn from a deck. Find the conditional probability of P(a queen ∣ a face card).

    (14.89)4/12
    Exercise 78.

    A card is drawn from a deck. Find the conditional probability of P(a queen ∣ a club).

    A die is rolled. Find the conditional probability that it shows a three if it is known that an odd number has shown.

    (14.90)1/3
    Exercise 80.

    If P(A)=.3 and P(B)=.4, and P(A   and  B)=.12, find the following.

    1. P(AB)

    2. P(BA)

    The following table shows the distribution of Democratic and Republican U.S. Senators by gender.

    Table 14.3.
     MALE(M)FEMALE(F)TOTAL
    DEMOCRATS(D)39443
    REPUBLICANS(R)52557
    TOTALS919100

    Use this table to determine the following probabilities:

    (14.91)P(MD)
    (14.92)39/43
    Exercise 82.
    (14.93)P(DM)
    (14.94)P(FR)
    5 / 57
    Exercise 84.
    (14.95)P(RF)

    Do the following conditional probability problems.

    At De Anza College, 20% of the students take Finite Math, 30% take History, and 5% take both Finite Math and History. If a student is chosen at random, find the following conditional probabilities.

    1. He is taking Finite Math given that he is taking History.

    2. He is taking History assuming that he is taking Finite Math.

    1. 1/6

    2. 1/4

    Exercise 86.

    At a college, 60% of the students pass Accounting, 70% pass English, and 30% pass both of these courses. If a student is selected at random, find the following conditional probabilities.

    1. He passes Accounting given that he passed English.

    2. He passes English assuming that he passed Accounting.

    If P(F)=.4 and P(EF)=.3, find P(E   and  F).

    0.12
    Exercise 88.

    If P(E)=.3, and P(F)=.3, and E and F are mutually exclusive, find P(EF).

    If P(E)=.6 and P(E  and   F)=.24, find P(FE).

    0.4
    Exercise 90.

    If P(E   and   F)=.04 and P(EF)=.1, find P(F).

    Consider a family of three children. Find the following probabilities.

    (14.96)P(two boys∣first born is a boy)
    (14.97)2/4
    Exercise 92.
    (14.98)P(all girls ∣ at least one girl is born)
    (14.99)P(children of both sexes ∣ first born is a boy)
    (14.100)3/4
    Exercise 94.
    (14.101)P(all boys ∣ there are children of both sexes)

    INDEPENDENT EVENTS

    The distribution of the number of fiction and non-fiction books checked out at a city's main library and at a smaller branch on a given day is as follows.

    Table 14.4.
     MAIN(M)BRANCH(B)TOTAL
    FICTION(F)300100400
    NON-FICTION(N)15050200
    TOTALS450150600

    Use this table to determine the following probabilities:

    (14.102)P(F)
    (14.103)2/3
    Exercise 96.
    (14.104)P(MF)
    (14.105)P(NB)
    (14.106)50/150
    Exercise 98.

    Is the fact that a person checks out a fiction book independent of the main library?

    For a two-child family, let the events E, F, and G be as follows.

    E: The family has at least one boy F: The family has children of both sexes G: The family's first born is a boy

    Find the following.

    1. P(E)

    2. P(F)

    3. P(EF)

    4. Are E and F independent?

    1. 3/4

    2. 2/4

    3. 2/4

    4. no

    Exercise 100.

    Find the following.

    1. P(F)

    2. P(G)

    3. P(FG)

    4. Are F and G independent?

    Do the following problems involving independence.

    If P(E)=.6, P(F)=.2, and E and F are independent, find P(E   and   F).

    0.12
    Exercise 102.

    If P(E)=.6, P(F)=.2, and E and F are independent, find P(E  or  F).

    If P(E)=.9, P(FE)=.36, and E and F are independent, find P(F).

    0.36
    Exercise 104.

    If P(E)=.6, P(E   or   F)=.08, and E and F are independent, find P(F).

    In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 6 had minor headaches. Of the non-drinkers, 9 had minor headaches. Are the events "drinkers" and "had headaches" independent?

    Yes
    Exercise 106.

    It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?

    John's probability of passing statistics is 40%, and Linda's probability of passing the same course is 70%. If the two events are independent, find the following probabilities.

    1. P(both of them will pass statistics)

    2. P(at least one of them will pass statistics)

    1. 28/100

    2. 82/100

    Exercise 108.

    Jane is flying home for the Christmas holidays. She has to change planes twice on the way home. There is an 80% chance that she will make the first connection, and a 90% chance that she will make the second connection. If the two events are independent, find the following probabilities.

    1. P(Jane will make both connections)

    2. P(Jane will make at least one connection)

    For a three-child family, let the events E, F, and G be as follows.

    E: The family has at least one boy F: The family has children of both sexes G: The family's first born is a boy

    Find the following.

    1. P(E)

    2. P(F)

    3. P(EF)

    4. Are E and F independent?

    1. 7/8

    2. 6/8

    3. 6/8

    4. no

    Exercise 110.

    Find the following.

    1. P(F)

    2. P(G)

    3. P(FG)

    4. Are F and G independent?

    CHAPTER REVIEW

    Two dice are rolled. Find the probability that the sum of the dice is

    1. four

    2. five

    1. 3/36

    2. 4/36

    A jar contains 3 red, 4 white, and 5 blue marbles. If a marble is chosen at random, find the following probabilities:

    1. P(red or blue)

    2. P(not blue)

    1. 8/12

    2. 7/12

    A card is drawn from a standard deck. Find the following probabilities:

    1. P(a jack or a king)

    2. P(a jack or a spade)

    1. 8/52

    2. 16/52

    A basket contains 3 red and 2 yellow apples. Two apples are chosen at random. Find the following probabilities:

    1. P(one red, one yellow)

    2. P(at least one red)

    1. 3/5

    2. 9/10

    A basket contains 4 red, 3 white, and 3 blue marbles. Three marbles are chosen at random. Find the following probabilities:

    1. P(two red, one white)

    2. P(first red, second white, third blue)

    3. P(at least one red)

    4. P(none red)

    1. 3/20

    2. 1/20

    3. 5/6

    4. 1/6

    Given a family of four children. Find the following probabilities:

    1. P(All boys)

    2. P(1 boy and 3 girls)

    1. 1/16

    2. 1/4

    Consider a family of three children. Find the following:

    1. P(children of both sexes ∣ first born is a boy)

    2. P(all girls ∣ children of both sexes)

    1. 3/4

    2. 0

    Mrs. Rossetti is flying from San Francisco to New York. On her way to the San Francisco Airport she encounters heavy traffic and determines that there is a 20% chance that she will be late to the airport and will miss her flight. Even if she makes her flight, there is a 10% chance that she will miss her connecting flight at Chicago. What is the probability that she will make it to New York as scheduled?

    0.72

    At a college, twenty percent of the students take history, thirty percent take math, and ten percent take both. What percent of the students take at least one of these two courses?

    40%

    In a T-maze, a mouse may run to the right (R) or may run to the left (L). A mouse goes up the maze three times, and events E and F are described as follows:

    E: Runs to the right on the first trial

    F: Runs to the left two consecutive times

    Determine whether the events E and F are independent.

    independent

    A college has found that 20% of its students take advanced math courses, 40% take advanced English courses and 15% take both advanced math and advanced English courses. If a student is selected at random, what is the probability that

    1. he is taking English given that he is taking math?

    2. he is taking math or English?

    1. 3/4

    2. 0.45

    If there are 35 students in a class, what is the probability that at least two have the same birthday?

    0.8144

    A student feels that her probability of passing accounting is .62, of passing mathematics is .45, and her passing accounting or mathematics is .85. Find the probability that the student passes both accounting and math.

    0.22

    There are nine judges on the U. S. Supreme Court of which five are conservative and four liberal. This year the court will act on six major cases. What is the probability that out of six cases the court will favor the conservatives in at least four?

    0.45278

    Five cards are drawn from a deck. Find the probability of obtaining

    1. four cards of a single suit

    2. two cards of one suit, two of another suit, and one from the remaining

    3. a pair(e.g. two aces and three other cards)

    4. a straight flush(five in a row of a single suit but not a royal flush)

    1. 111540/2598960

    2. 949104/2598960

    3. 1349088/2598960

    4. 36/2598960

    The following table shows a distribution of drink preferences by gender.

    Table 14.5.
     Coke(C)Pepsi(P)Seven Up(S)TOTALS
    Males(M)605022132
    Females(F)504018108
    TOTALS1109040240

    The events M, F, C, P and S are defined as Male, Female, Coca Cola, Pepsi, and Seven Up, respectively. Find the following:

    1. P(FS)

    2. P(PF)

    3. P(CM)

    4. _autogen-svg2png-0274.png

    5. Are the events F and S mutually exclusive?

    6. Are the events F and S independent?

    1. 9/20

    2. 10/27

    3. 15/33

    4. 11/20

    5. no

    6. yes

    At a clothing outlet 20% of the clothes are irregular, 10% have at least a button missing and 4% are both irregular and have a button missing. If Martha found a dress that has a button missing, what is the probability that it is irregular?

    0.40

    A trade delegation consists of four Americans, three Japanese and two Germans. Three people are chosen at random. Find the following probabilities:

    1. P(two Americans and one Japanese)

    2. P(at least one American)

    3. P(One of each nationality)

    4. P(no German)

    1. 3/14

    2. 37/42

    3. 2/7

    4. 35/84

    A coin is tossed three times, and the events E and F are as follows.

    E: It shows a head on the first toss F: Never turns up a tail

    Are the events E and F independent?

    No

    If P(E)=.6 and P(F)=.4 and E and F are mutually exclusive, find P(E   and   F).

    0

    If P(E)=.5 and P(F)=.3 and E and F are independent, find _autogen-svg2png-0306.png.

    0.65

    If P(F)=.9 and P(EF)=.36 and E and F are independent, find P(E).

    0.36

    If P(E)=.4 and P(E  or  F)=.9 and E and F are independent, find P(F).

    (14.107)5/6

    If P(E)=.4 and P(FE)=.5, find P(E   and  F).

    0.2

    If P(E)=.6 and P(E  and  F)=.3, find P(FE).

    0.5

    If P(E)=.3 and P(F)=.4 and E and F are independent, find P(EF).

    0.3
    Solutions