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Probability: Homework

Module by: UniqU, LLC. E-mail the author

Based on: Applied Finite Mathematics: Chapter 07 by Rupinder Sekhon

Summary: This chapter covers principles of probability. After completing this chapter students should be able to: write sample spaces; determine whether two events are mutually exclusive; use the addition rule; calculate probabilities using tree diagrams and combinations; solve problems involving conditional probability; determine whether two events are independent.

SAMPLE SPACES AND PROBABILITY

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

Exercise 1

A die is rolled.

Solution

1,2,3,4,5,61,2,3,4,5,6 size 12{ left lbrace 1,2,3,4,5,6 right rbrace } {}
(1)

Exercise 2

A penny and a nickel are tossed.

Exercise 3

A die is rolled, and a coin is tossed.

Solution

1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T size 12{ left lbrace 1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T right rbrace } {}
(2)

Exercise 4

Three coins are tossed.

Exercise 5

Two dice are rolled.

Solution

Table 1
  1 2 3 4 5 6
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.

Exercise 7

Pan acePan ace size 12{P left ("an ace" right )} {}
(3)

Solution

4/524/52 size 12{4/"52"} {}
(4)

Exercise 8

Pa red cardPa red card size 12{P left ("a red card" right )} {}
(5)

Exercise 9

Pa clubPa club size 12{P left ("a club" right )} {}
(6)

Solution

13/5213/52 size 12{"13"/"52"} {}
(7)

Exercise 10

Pa face cardPa face card size 12{P left ("a face card" right )} {}
(8)

Exercise 11

Pa jack or spadePa jack or spade size 12{P left ("a jack or spade" right )} {}
(9)

Solution

16/5216/52 size 12{"16"/"52"} {}
(10)

Exercise 12

Pa jack and a spadePa jack and a spade size 12{P left ("a jack and a spade" right )} {}
(11)

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

Exercise 13

PredPred size 12{P left ("red" right )} {}
(12)

Solution

6/206/20 size 12{6/"20"} {}
(13)

Exercise 14

PwhitePwhite size 12{P left ("white" right )} {}
(14)

Exercise 15

Pred or bluePred or blue size 12{P left ("red or blue" right )} {}
(15)

Solution

13/2013/20 size 12{"13"/"20"} {}
(16)

Exercise 16

Pred and bluePred and blue size 12{P left ("red and blue" right )} {}
(17)

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

Exercise 17

Ptwo boys and a girlPtwo boys and a girl size 12{P left ("two boys and a girl" right )} {}
(18)

Solution

3/83/8 size 12{3/8} {}
(19)

Exercise 18

Pat least one boyPat least one boy size 12{P left ("at least one boy" right )} {}
(20)

Exercise 19

Pchildren of both sexesPchildren of both sexes size 12{P left ("children of both sexes" right )} {}
(21)

Solution

6/86/8 size 12{6/8} {}
(22)

Exercise 20

Pat most one girlPat most one girl size 12{P left ("at most one girl" right )} {}
(23)

Two dice are rolled. Find the following probabilities.

Exercise 21

Pthe sum of the dice is 5Pthe sum of the dice is 5 size 12{P left ("the sum of the dice is 5" right )} {}
(24)

Solution

4/364/36 size 12{4/"36"} {}
(25)

Exercise 22

Pthe sum of the dice is 8Pthe sum of the dice is 8 size 12{P left ("the sum of the dice is 8" right )} {}
(26)

Exercise 23

Pthe sum is 3 or 6Pthe sum is 3 or 6 size 12{P left ("the sum is 3 or 6" right )} {}
(27)

Solution

7/367/36 size 12{7/"36"} {}
(28)

Exercise 24

Pthe sum is more than 10Pthe sum is more than 10 size 12{P left ("the sum is more than 10" right )} {}
(29)

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

Exercise 25

Pthe sum of the number is 5Pthe sum of the number is 5 size 12{P left ("the sum of the number is 5" right )} {}
(30)

Solution

4/124/12 size 12{4/"12"} {}
(31)

Exercise 26

Pthe sum of the numbers is oddPthe sum of the numbers is odd size 12{P left ("the sum of the numbers is odd" right )} {}
(32)

Exercise 27

Pthe sum of the numbers is 9Pthe sum of the numbers is 9 size 12{P left ("the sum of the numbers is 9" right )} {}
(33)

Solution

0

Exercise 28

Pone of the numbers is 3Pone of the numbers is 3 size 12{P left ("one of the numbers is 3" right )} {}
(34)

MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE

Determine whether the following pair of events are mutually exclusive.

Exercise 29

A=A person earns more than $25,000A=A person earns more than $25,000 size 12{A= left lbrace "A person earns more than $25,000" right rbrace } {}
(35)
B=A person earns less than $20,000B=A person earns less than $20,000 size 12{B= left lbrace "A person earns less than $20,000" right rbrace } {}
(36)

Solution

Yes

Exercise 30

A card is drawn from a deck.

C=It is a KingD=It is a heart.C=It is a King size 12{C= left lbrace "It is a King" right rbrace } {}D=It is a heart size 12{D= left lbrace "It is a heart" right rbrace } {}.
(37)

Exercise 31

A die is rolled.

E=An even number showsE=An even number shows size 12{E= left lbrace "An even number shows" right rbrace } {}
(38)
F=A number greater than 3 showsF=A number greater than 3 shows size 12{F= left lbrace "A number greater than 3 shows" right rbrace } {}
(39)

Solution

No

Exercise 32

Two dice are rolled.

G=The sum of dice is 8G=The sum of dice is 8 size 12{G= left lbrace "The sum of dice is 8" right rbrace } {}
(40)
H=One die shows a 6H=One die shows a 6 size 12{H= left lbrace "One die shows a 6" right rbrace } {}
(41)

Exercise 33

Three coins are tossed.

I=Two heads come upI=Two heads come up size 12{I= left lbrace "Two heads come up" right rbrace } {}
(42)
J=At least one tail comes upJ=At least one tail comes up size 12{J= left lbrace "At least one tail comes up" right rbrace } {}
(43)

Solution

No

Exercise 34

A family has three children.

K=First born is a boyK=First born is a boy size 12{K= left lbrace "First born is a boy" right rbrace } {}
(44)
L=The family has children of both sexesL=The family has children of both sexes size 12{L= left lbrace "The family has children of both sexes" right rbrace } {}
(45)

Use the addition rule to find the following probabilities.

Exercise 35

A card is drawn from a deck, and the events CC size 12{C} {} and DD size 12{D} {} are as follows:

C=It is a kingC=It is a king size 12{C= left lbrace "It is a king" right rbrace } {}
(46)
D=It is a heartD=It is a heart size 12{D= left lbrace "It is a heart" right rbrace } {}
(47)

Find PC  or   DPC  or   D size 12{P left (C" or "D right )} {}.

Solution

16/5216/52 size 12{"16"/"52"} {}
(48)

Exercise 36

A die is rolled, and the events EE size 12{E} {} and FF size 12{F} {} are as follows:

E=An even number showsE=An even number shows size 12{E= left lbrace "An even number shows" right rbrace } {}
(49)
F=A number greater than 3 showsF=A number greater than 3 shows size 12{F= left lbrace "A number greater than 3 shows" right rbrace } {}
(50)

Find PE   or   FPE   or   F size 12{P left (E" or "F right )} {}.

Exercise 37

Two dice are rolled, and the events GG size 12{G} {} and HH size 12{H} {} are as follows:

G=The sum of dice is 8G=The sum of dice is 8 size 12{G= left lbrace "The sum of dice is 8" right rbrace } {}
(51)
H=Exactly one die shows a 6H=Exactly one die shows a 6 size 12{H= left lbrace "Exactly one die shows a 6" right rbrace } {}
(52)

Find PG  or   HPG  or   H size 12{P left (G" or "H right )} {}.

Solution

13/3613/36 size 12{"13"/"36"} {}
(53)

Exercise 38

Three coins are tossed, and the events II size 12{I} {} and JJ size 12{J} {} are as follows:

I=Two heads come upI=Two heads come up size 12{I= left lbrace "Two heads come up" right rbrace } {}
(54)
J=At least one tail comes upJ=At least one tail comes up size 12{J= left lbrace "At least one tail comes up" right rbrace } {}
(55)

Find PI   or  JPI   or  J size 12{P left (I" or "J right )} {}.

Exercise 39

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?

Solution

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 2
  MALES(M) FEMALES(F) TOTAL
DEMOCRATS(D) 39 4 43
REPUBLICANS(R) 52 5 57
TOTALS 91 9 100

Use this table to determine the following probabilities.

Exercise 41

PM   and  DPM   and  D size 12{P left (M" and "D right )} {}
(56)

Solution

39/10039/100 size 12{"39"/"100"} {}
(57)

Exercise 42

PF   and   RPF   and   R size 12{P left (F" and "R right )} {}
(58)

Exercise 43

PM   or  DPM   or  D size 12{P left (M" or "D right )} {}
(59)

Solution

95/10095/100 size 12{"95"/"100"} {}
(60)

Exercise 44

PF   or   RCPF   or   RC size 12{P left (F" or "R rSup { size 8{C} } right )} {}
(61)

Exercise 45

PMC   or  RPMC   or  R size 12{P left (M rSup { size 8{C} } " or "R right )} {}
(62)

Solution

61/10061/100 size 12{"61"/"100"} {}
(63)

Exercise 46

PM  or   FPM  or   F size 12{P left (M" or "F right )} {}
(64)

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

Exercise 47

If PE=.5PE=.5 size 12{P left (E right )= "." 5} {} and PF=.4PF=.4 size 12{P left (F right )= "." 4} {} and EE size 12{E} {} and FF size 12{F} {} are mutually exclusive, find PE   and   FPE   and   F size 12{P left (E" and "F right )} {}.

Solution

0

Exercise 48

If PE=.4PE=.4 size 12{P left (E right )= "." 4} {} and PF=.2PF=.2 size 12{P left (F right )= "." 2} {} and EE size 12{E} {} and FF size 12{F} {} are mutually exclusive, find PE  or  FPE  or  F size 12{P left (E" or "F right )} {}.

Exercise 49

If PE=.3PE=.3 size 12{P left (E right )= "." 3} {} and PE   or   F=.6PE   or   F=.6 size 12{P left (E" or "F right )= "." 6} {} and PE   and   F=.2PE   and   F=.2 size 12{P left (E" and "F right )= "." 2} {}, find PFPF size 12{P left (F right )} {}.

Solution

0.5

Exercise 50

If PE=.4PE=.4 size 12{P left (E right )= "." 4} {}, PF=.5PF=.5 size 12{P left (F right )= "." 5} {} and PE  or  F=.7PE  or  F=.7 size 12{P left (E" or "F right )= "." 7} {}, find PE   and  FPE   and  F size 12{P left (E" and "F right )} {}.

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.

Exercise 51

Pboth redPboth red size 12{P left ("both red" right )} {}
(65)

Solution

20/5620/56 size 12{"20"/"56"} {}
(66)

Exercise 52

Pone red, one yellowPone red, one yellow size 12{P left ("one red, one yellow" right )} {}
(67)

Exercise 53

Pboth yellowPboth yellow size 12{P left ("both yellow" right )} {}
(68)

Solution

6/566/56 size 12{6/"56"} {}
(69)

Exercise 54

PFirst red and second yellowPFirst red and second yellow size 12{P left ("First red and second yellow" right )} {}
(70)

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

Exercise 55

PAll three redPAll three red size 12{P left ("All three red" right )} {}
(71)

Solution

1/61/6 size 12{1/6} {}
(72)

Exercise 56

Ptwo red, one bluePtwo red, one blue size 12{P left ("two red, one blue" right )} {}
(73)

Exercise 57

Pone red, two bluePone red, two blue size 12{P left ("one red, two blue" right )} {}
(74)

Solution

3/103/10 size 12{3/"10"} {}
(75)

Exercise 58

Pfirst red, second blue, third redPfirst red, second blue, third red size 12{P left ("first red, second blue, third red" right )} {}
(76)

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

Exercise 59

Pall three redPall three red size 12{P left ("all three red" right )} {}
(77)

Solution

10/22010/220 size 12{"10"/"220"} {}
(78)

Exercise 60

Ptwo white and 1 bluePtwo white and 1 blue size 12{P left ("two white and 1 blue" right )} {}
(79)

Exercise 61

Pnone whitePnone white size 12{P left ("none white" right )} {}
(80)

Solution

56/22056/220 size 12{"56"/"220"} {}
(81)

Exercise 62

Pat least one redPat least one red size 12{P left ("at least one red" right )} {}
(82)

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

Exercise 63

At least three freshmen.

Solution

45/136545/1365 size 12{"45"/"1365"} {}
(83)

Exercise 64

No sophomores.

Exercise 65

All four of the same class.

Solution

21/136521/1365 size 12{"21"/"1365"} {}
(84)

Exercise 66

Not all four from the same class.

Exercise 67

Exactly three of the same class.

Solution

324/1365324/1365 size 12{"324"/"1365"} {}
(85)

Exercise 68

More juniors than freshmen and sophomores combined.

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

Exercise 69

Two hearts, two spades, and one club.

Solution

79092/2,598,96079092/2,598,960 size 12{"79092"/2,"598","960"} {}
(86)

Exercise 70

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

Exercise 71

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

Solution

24/2,598,96024/2,598,960 size 12{"24"/2,"598","960"} {}
(87)

Exercise 72

Any full house.

Exercise 73

A pair of nines and tens.

Solution

1,584/2,598,9601,584/2,598,960 size 12{1,"584"/2,"598","960"} {}
(88)

Exercise 74

Two pairs

Do the following birthday problems.

Exercise 75

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

Solution

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: PA∣B=PA∩BPBPA∣B=PA∩BPB size 12{P left (A \lline B right )= { {P left (A intersection B right )} over {P left (B right )} } } {}.

Exercise 77

A card is drawn from a deck. Find the conditional probability of Pa queen ∣ a face cardPa queen ∣ a face card size 12{P left ("a queen " \lline " a face card" right )} {}.

Solution

4/124/12 size 12{4/"12"} {}
(89)

Exercise 78

A card is drawn from a deck. Find the conditional probability of Pa queen ∣ a clubPa queen ∣ a club size 12{P left ("a queen " \lline " a club" right )} {}.

Exercise 79

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

Solution

1/31/3 size 12{1/3} {}
(90)

Exercise 80

If PA=.3PA=.3 size 12{P left (A right )= "." 3} {} and PB=.4PB=.4 size 12{P left (B right )= "." 4} {}, and PA   and  B=.12PA   and  B=.12 size 12{P left (A" and "B right )= "." "12"} {}, find the following.

  1. PA∣BPA∣B size 12{P left (A \lline B right )} {}

  2. PB∣APB∣A size 12{P left (B \lline A right )} {}

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

Table 3
  MALE(M) FEMALE(F) TOTAL
DEMOCRATS(D) 39 4 43
REPUBLICANS(R) 52 5 57
TOTALS 91 9 100

Use this table to determine the following probabilities:

Exercise 81

PM∣DPM∣D size 12{P left (M \lline D right )} {}
(91)

Solution

39/4339/43 size 12{"39"/"43"} {}
(92)

Exercise 82

PD∣MPD∣M size 12{P left (D \lline M right )} {}
(93)

Exercise 83

PF∣RPF∣R size 12{P left (F \lline R right )} {}
(94)

Solution

5 / 57 5/57

Exercise 84

PR∣FPR∣F size 12{P left (R \lline F right )} {}
(95)

Do the following conditional probability problems.

Exercise 85

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.

Solution

  1. 1/61/6 size 12{1/6} {}
  2. 1/41/4 size 12{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.

Exercise 87

If PF=.4PF=.4 size 12{P left (F right )= "." 4} {} and PE∣F=.3PE∣F=.3 size 12{P left (E \lline F right )= "." 3} {}, find PE   and  FPE   and  F size 12{P left (E" and "F right )} {}.

Solution

0.12

Exercise 88

If PE=.3PE=.3 size 12{P left (E right )= "." 3} {}, and PF=.3PF=.3 size 12{P left (F right )= "." 3} {}, and EE size 12{E} {} and FF size 12{F} {} are mutually exclusive, find PE∣FPE∣F size 12{P left (E \lline F right )} {}.

Exercise 89

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {} and PE  and   F=.24PE  and   F=.24 size 12{P left (E" and "F right )= "." "24"} {}, find PF∣EPF∣E size 12{P left (F \lline E right )} {}.

Solution

0.4

Exercise 90

If PE   and   F=.04PE   and   F=.04 size 12{P left (E" and "F right )= "." "04"} {} and PE∣F=.1PE∣F=.1 size 12{P left (E \lline F right )= "." 1} {}, find PFPF size 12{P left (F right )} {}.

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

Exercise 91

Ptwo boys∣first born is a boyPtwo boys∣first born is a boy size 12{P left ("two boys" \lline "first born is a boy" right )} {}
(96)

Solution

2/42/4 size 12{2/4} {}
(97)

Exercise 92

Pall girls ∣ at least one girl is bornPall girls ∣ at least one girl is born size 12{P left ("all girls " \lline " at least one girl is born" right )} {}
(98)

Exercise 93

Pchildren of both sexes ∣ first born is a boyPchildren of both sexes ∣ first born is a boy size 12{P left ("children of both sexes " \lline " first born is a boy" right )} {}
(99)

Solution

3/43/4 size 12{3/4} {}
(100)

Exercise 94

Pall boys ∣ there are children of both sexesPall boys ∣ there are children of both sexes size 12{P left ("all boys " \lline " there are children of both sexes" right )} {}
(101)

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 4
  MAIN(M) BRANCH(B) TOTAL
FICTION(F) 300 100 400
NON-FICTION(N) 150 50 200
TOTALS 450 150 600

Use this table to determine the following probabilities:

Exercise 95

PFPF size 12{P left (F right )} {}
(102)

Solution

2/32/3 size 12{2/3} {}
(103)

Exercise 96

PM∣FPM∣F size 12{P left (M \lline F right )} {}
(104)

Exercise 97

PN∣BPN∣B size 12{P left (N \lline B right )} {}
(105)

Solution

50/15050/150 size 12{"50"/"150"} {}
(106)

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 EE size 12{E} {}, FF size 12{F} {}, and GG size 12{G} {} be as follows.

EE size 12{E} {}: The family has at least one boy FF size 12{F} {}: The family has children of both sexes GG size 12{G} {}: The family's first born is a boy

Exercise 99

Find the following.

  1. PEPE size 12{P left (E right )} {}

  2. PFPF size 12{P left (F right )} {}

  3. PE∩FPE∩F size 12{P left (E intersection F right )} {}

  4. Are EE size 12{E} {} and FF size 12{F} {} independent?

Solution

  1. 3/43/4 size 12{3/4} {}
  2. 2/42/4 size 12{2/4} {}
  3. 2/42/4 size 12{2/4} {}
  4. no

Exercise 100

Find the following.

  1. PFPF size 12{P left (F right )} {}

  2. PGPG size 12{P left (G right )} {}

  3. PF∩GPF∩G size 12{P left (F intersection G right )} {}

  4. Are FF size 12{F} {} and GG size 12{G} {} independent?

Do the following problems involving independence.

Exercise 101

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {}, PF=.2PF=.2 size 12{P left (F right )= "." 2} {}, and EE size 12{E} {} and FF size 12{F} {} are independent, find PE   and   FPE   and   F size 12{P left (E" and "F right )} {}.

Solution

0.12

Exercise 102

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {}, PF=.2PF=.2 size 12{P left (F right )= "." 2} {}, and EE size 12{E} {} and FF size 12{F} {} are independent, find PE  or  FPE  or  F size 12{P left (E" or "F right )} {}.

Exercise 103

If PE=.9PE=.9 size 12{P left (E right )= "." 9} {}, PF∣E=.36PF∣E=.36 size 12{P left (F \lline E right )= "." "36"} {}, and EE size 12{E} {} and FF size 12{F} {} are independent, find PFPF size 12{P left (F right )} {}.

Solution

0.36

Exercise 104

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {}, PE   or   F=.08PE   or   F=.08 size 12{P left (E" or "F right )= "." "08"} {}, and EE size 12{E} {} and FF size 12{F} {} are independent, find PFPF size 12{P left (F right )} {}.

Exercise 105

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?

Solution

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?

Exercise 107

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. Pboth of them will pass statisticsPboth of them will pass statistics size 12{P left ("both of them will pass statistics" right )} {}

  2. Pat least one of them will pass statisticsPat least one of them will pass statistics size 12{P left ("at least one of them will pass statistics" right )} {}

Solution

  1. 28/10028/100 size 12{"28"/"100"} {}
  2. 82/10082/100 size 12{"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. PJane will make both connectionsPJane will make both connections size 12{P left ("Jane will make both connections" right )} {}

  2. PJane will make at least one connectionPJane will make at least one connection size 12{P left ("Jane will make at least one connection" right )} {}

For a three-child family, let the events EE size 12{E} {}, FF size 12{F} {}, and GG size 12{G} {} be as follows.

EE size 12{E} {}: The family has at least one boy FF size 12{F} {}: The family has children of both sexes GG size 12{G} {}: The family's first born is a boy

Exercise 109

Find the following.

  1. PEPE size 12{P left (E right )} {}

  2. PFPF size 12{P left (F right )} {}

  3. PE∩FPE∩F size 12{P left (E intersection F right )} {}

  4. Are EE size 12{E} {} and FF size 12{F} {} independent?

Solution

  1. 7/87/8 size 12{7/8} {}
  2. 6/86/8 size 12{6/8} {}
  3. 6/86/8 size 12{6/8} {}
  4. no

Exercise 110

Find the following.

  1. PFPF size 12{P left (F right )} {}

  2. PGPG size 12{P left (G right )} {}

  3. PF∩GPF∩G size 12{P left (F intersection G right )} {}

  4. Are FF size 12{F} {} and GG size 12{G} {} independent?

CHAPTER REVIEW

Exercise 111

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

  1. four
  2. five

Solution

  1. 3/363/36 size 12{3/"36"} {}
  2. 4/364/36 size 12{4/"36"} {}

Exercise 112

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

  1. Pred or bluePred or blue size 12{P left ("red or blue" right )} {}
  2. Pnot bluePnot blue size 12{P left ("not blue" right )} {}

Solution

  1. 8/128/12 size 12{8/"12"} {}
  2. 7/127/12 size 12{7/"12"} {}

Exercise 113

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

  1. Pa jack or a kingPa jack or a king size 12{P left ("a jack or a king" right )} {}
  2. Pa jack or a spadePa jack or a spade size 12{P left ("a jack or a spade" right )} {}

Solution

  1. 8/528/52 size 12{8/"52"} {}
  2. 16/5216/52 size 12{"16"/"52"} {}

Exercise 114

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

  1. Pone red, one yellowPone red, one yellow size 12{P left ("one red, one yellow" right )} {}
  2. Pat least one redPat least one red size 12{P left ("at least one red" right )} {}

Solution

  1. 3/53/5 size 12{3/5} {}
  2. 9/109/10 size 12{9/"10"} {}

Exercise 115

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

  1. Ptwo red, one whitePtwo red, one white size 12{P left ("two red, one white" right )} {}
  2. Pfirst red, second white, third bluePfirst red, second white, third blue size 12{P left ("first red, second white, third blue" right )} {}
  3. Pat least one redPat least one red size 12{P left ("at least one red" right )} {}
  4. Pnone redPnone red size 12{P left ("none red" right )} {}

Solution

  1. 3/203/20 size 12{3/"20"} {}
  2. 1/201/20 size 12{1/"20"} {}
  3. 5/65/6 size 12{5/6} {}
  4. 1/61/6 size 12{1/6} {}

Exercise 116

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

  1. PAll boysPAll boys size 12{P left ("All boys" right )} {}
  2. P1 boy and 3 girlsP1 boy and 3 girls size 12{P left ("1 boy and 3 girls" right )} {}

Solution

  1. 1/161/16 size 12{1/"16"} {}
  2. 1/41/4 size 12{1/4} {}

Exercise 117

Consider a family of three children. Find the following:

  1. Pchildren of both sexes ∣ first born is a boyPchildren of both sexes ∣ first born is a boy size 12{P left ("children of both sexes " \lline " first born is a boy" right )} {}
  2. Pall girls ∣ children of both sexesPall girls ∣ children of both sexes size 12{P left ("all girls " \lline " children of both sexes" right )} {}

Solution

  1. 3/43/4 size 12{3/4} {}
  2. 0

Exercise 118

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?

Solution

0.72

Exercise 119

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?

Solution

40%

Exercise 120

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 EE size 12{E} {} and FF size 12{F} {} are described as follows:

EE size 12{E} {}: Runs to the right on the first trial

FF size 12{F} {}: Runs to the left two consecutive times

Determine whether the events EE size 12{E} {} and FF size 12{F} {} are independent.

Solution

independent

Exercise 121

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?

Solution

  1. 3/43/4 size 12{3/4} {}
  2. 0.45

Exercise 122

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

Solution

0.8144

Exercise 123

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.

Solution

0.22

Exercise 124

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?

Solution

0.45278

Exercise 125

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)

Solution

  1. 111540/2598960111540/2598960 size 12{"111540"/"2598960"} {}
  2. 949104/2598960949104/2598960 size 12{"949104"/"2598960"} {}
  3. 1349088/25989601349088/2598960 size 12{"1349088"/"2598960"} {}
  4. 36/259896036/2598960 size 12{"36"/"2598960"} {}

Exercise 126

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

Table 5
  Coke(C) Pepsi(P) Seven Up(S) TOTALS
Males(M) 60 50 22 132
Females(F) 50 40 18 108
TOTALS 110 90 40 240

The events MM size 12{M} {}, FF size 12{F} {}, CC size 12{C} {}, PP size 12{P} {} and SS size 12{S} {} are defined as Male, Female, Coca Cola, Pepsi, and Seven Up, respectively. Find the following:

  1. PF∣SPF∣S size 12{P left (F \lline S right )} {}
  2. PP∣FPP∣F size 12{P left (P \lline F right )} {}
  3. PC∣MPC∣M size 12{P left (C \lline M right )} {}
  4. PM∣P⋃ CPM∣P⋃ C size 12{P left (M \lline ital "PUC" right )} {}
  5. Are the events FF size 12{F} {} and SS size 12{S} {} mutually exclusive?
  6. Are the events FF size 12{F} {} and SS size 12{S} {} independent?

Solution

  1. 9/209/20 size 12{9/"20"} {}
  2. 10/2710/27 size 12{"10"/"27"} {}
  3. 15/3315/33 size 12{"15"/"33"} {}
  4. 11/2011/20 size 12{"11"/"20"} {}
  5. no
  6. yes

Exercise 127

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?

Solution

0.40

Exercise 128

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

  1. Ptwo Americans and one JapanesePtwo Americans and one Japanese size 12{P left ("two Americans and one Japanese" right )} {}
  2. Pat least one AmericanPat least one American size 12{P left ("at least one American" right )} {}
  3. POne of each nationalityPOne of each nationality size 12{P left ("One of each nationality" right )} {}
  4. Pno GermanPno German size 12{P left ("no German" right )} {}

Solution

  1. 3/143/14 size 12{3/"14"} {}
  2. 37/4237/42 size 12{"37"/"42"} {}
  3. 2/72/7 size 12{2/7} {}
  4. 35/8435/84 size 12{"35"/"84"} {}

Exercise 129

A coin is tossed three times, and the events EE size 12{E} {} and FF size 12{F} {} are as follows.

EE size 12{E} {}: It shows a head on the first toss FF size 12{F} {}: Never turns up a tail

Are the events EE size 12{E} {} and FF size 12{F} {} independent?

Solution

No

Exercise 130

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {} and PF=.4PF=.4 size 12{P left (F right )= "." 4} {} and EE size 12{E} {} and FF size 12{F} {} are mutually exclusive, find PE   and   FPE   and   F size 12{P left (E" and "F right )} {}.

Solution

0

Exercise 131

If PE=.5PE=.5 size 12{P left (E right )= "." 5} {} and PF=.3PF=.3 size 12{P left (F right )= "." 3} {} and EE size 12{E} {} and FF size 12{F} {} are independent, find PE ⋃ FPE ⋃ F size 12{P left ( ital "EUF" right )} {}.

Solution

0.65

Exercise 132

If PF=.9PF=.9 size 12{P left (F right )= "." 9} {} and PE∣F=.36PE∣F=.36 size 12{P left (E \lline F right )= "." "36"} {} and EE size 12{E} {} and FF size 12{F} {} are independent, find PEPE size 12{P left (E right )} {}.

Solution

0.36

Exercise 133

If PE=.4PE=.4 size 12{P left (E right )= "." 4} {} and PE  or  F=.9PE  or  F=.9 size 12{P left (E" or "F right )= "." 9} {} and EE size 12{E} {} and FF size 12{F} {} are independent, find PFPF size 12{P left (F right )} {}.

Solution

5/65/6 size 12{5/6} {}
(107)

Exercise 134

If PE=.4PE=.4 size 12{P left (E right )= "." 4} {} and PF∣E=.5PF∣E=.5 size 12{P left (F \lline E right )= "." 5} {}, find PE   and  FPE   and  F size 12{P left (E" and "F right )} {}.

Solution

0.2

Exercise 135

If PE=.6PE=.6 size 12{P left (E right )= "." 6} {} and PE  and  F=.3PE  and  F=.3 size 12{P left (E" and "F right )= "." 3} {}, find PF∣EPF∣E size 12{P left (F \lline E right )} {}.

Solution

0.5

Exercise 136

If PE=.3PE=.3 size 12{P left (E right )= "." 3} {} and PF=.4PF=.4 size 12{P left (F right )= "." 4} {} and EE size 12{E} {} and FF size 12{F} {} are independent, find PE∣FPE∣F size 12{P left (E \lline F right )} {}.

Solution

0.3

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