Chapter 4 - Probability and Counting Rules

Note: Answers may vary due to rounding, 13.

TI-83's or computer programs. There are 6# or 36 outcomes.
a. There are 4 ways to get a sum of 9. They
EXERCISE SET 4-1 are (6,3), (5,4), (4,5), and (3,6). The
%
probability then is $' œ "* Þ
1.
A probability experiment is a chance process b. There are 6 ways to get a sum of 7 and 2
which leads to well-defined outcomes. ways to get a sum of 11. They are (6,1),
(5,2), (4,3), (3,4), (2,5), (1,6), (6,5) and
)
2. (5,6). The probability then is $' œ #* Þ
The set of all possible outcomes of a
probability experiment is called a sample c. There are 6 ways to get doubles. They
space. are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).
'
The probability then is $' œ "' Þ
3.
An outcome is the result of a single trial of a d. To get a sum less than nine, one must roll
probability experiment, whereas an event a 2, 3, 4, 5, 6, 7, or 8. There are 26 ways to
can consist of one or more outcomes. get a sum less than 9. The probability then
is #' "$
$' œ ") Þ
4.
Equally likely events have the same
e. To get a sum greater than or equal to 10,
probability of occurring.
one must roll a 10, 11, or 12. There are six
ways to do this. They are (6,4), (5,5), (4,6),
5.
(6,5), (5,6), and (6,6). The probability is
The range of values is 0 Ÿ P(E) Ÿ 1. ' "
$' œ ' Þ

6.
14.
1 " "
a. "$ f. &#
7. " "
0 b. %
g. #'

" "
8. c. &#
h. #'
1
# "
d. "$
i. #
9.
1  0.20 œ 0.80 % "
e. "$
j. #
Since the probability that it won't rain is
80%, you could leave your umbrella at home 15.
and be fairly safe. There are 20 possible outcomes.

10. a. P(winning $10) œ P(rolling a 1)

c, d, e, h # "
P(rolling a 1) œ #! œ "! œ !Þ"

11. b. P(winning $5 or $10) œ P(rolling either

a. Empirical e. Empirical a 1or 2)
b. Classical f. Empirical %
P(1or 2) œ #! œ "& œ !Þ#
c. Empirical g. Subjective
d. Classical c. P(winning a coupon) œ P(rolling either a
3 or 4)
12. P(3 or 4) œ "' œ %& œ !Þ)
a. "' d. #$ g. "
$
#!

b. 0 e. 1 16.
c. "# f. #$ a. P(begins with M) œ )
œ %
&! #&

62
 Chapter 4 - Probability and Counting Rules

16. continued 23.

b. P(begins with a vowel) œ "#
&!
œ '
#&
The outcomes for 2, 3, or 12 are (1,1), (1,2),
'
P(not a vowel) œ "  #& œ "*
#&
(2,1), and (6,6); hence P(2, 3, or 12) œ
"#" %
$' œ $' œ *" Þ
17.
a. P(type O) œ 0.43 24.
#%ß&!"
a. P(50 or fewer) œ )$ß!&(
œ 0.295
b. P(type A or B) œ 0.40  0.12 œ 0.52
$%ß)!$
b. P(more than 100) œ )$ß!&(
œ !Þ419
c. P(not type A or O) œ "  0.83 œ 0.17
(('!
c. P(no more than 20) œ )$ß!&(
œ 0.093
18.
P(male) œ "  P(female)
P(male) œ 1  0.572 œ 0.428 25.
a. P(debt is less than $5001) œ 27%Þ
19.
a. P(even prime number) œ "
œ !Þ04 b. P(debt is more than $20,000) œ
#&
P($20,001 to $50,000) + P($50,000+) œ
"$ 19%  14% œ 33%
b. P(sum of the digits is even) œ #&
œ !Þ52

"! c. P(debt is between $1 and $20,000) œ

c. P(greater than 50) œ œ !Þ4
#& P($1 to $5000)  P($5001 to $20,000) œ
27%  40% œ 67%
20.
"*
a. P(60 or 70 mph) œ &!
œ !Þ38 d. P(debt is more than $50,000) œ 14%
$"
b. P(greater than 65 mph) œ &!
œ !Þ62 26.
(
a. P(Japanese) œ "!
œ 0.7
$(
c. P(70 mph or less) œ &!
œ !Þ74
"!
b. P(Japanese or German) œ "!
œ1
21.
The sample space is BBB, BBG, BGB, c. P(not foreign) œ !
œ0
"!
GBB, GGB, GBG, BGG, and GGG.
27.
a. All boys is the outcome BBB; hence P(ten thousand dollar bill) œ $ß%'!ß!!!
œ 0.662
P(all boys) œ ") Þ &ß##&ß!!!

28.
b. All girls or all boys would be BBB and a. P(not oil) œ 1  0.39 œ 0.61
GGG; hence, P(all girls or all boys) œ "% Þ
b. P(natural gas or oil) œ
c. Exactly two boys or two girls would be 0.39  0.24 œ 0.63
BBG, BGB, GBB, BBG, GBG, or BGG.
The probability then is ') œ $% Þ c. P(nuclear) œ 0.08

d. At least one child of each gender means 29.

at least one boy or at least one girl. The (a)
outcomes are the same as those of part c, 1 2 3 4 5 6
hence the probability is the same, $% Þ 1 1 2 3 4 5 6
2 2 4 6 8 10 12
22. 3 3 6 9 12 15 18
There are 6 ways to get a 7 and 2 ways to get 4 4 8 12 16 20 24
)
11; hence, P(7 or 11) œ $' œ #* Þ 5 5 10 15 20 25 30
6 6 12 18 24 30 36

63
 Chapter 4 - Probability and Counting Rules

29. continued 33.

"& & 1 1,1
(b) P(multiple of 6) œ $'
œ "#
2 1,2
1 3 1,3
"(
(c) P(less than 10) œ $' 4 1,4

1 2,1
30.
2 2,2
P(individual or corporate taxes) œ 0.60 2 3 2,3
4 2,4
31.
$5 $1, $5 1 3,1
$1 $10 $1, $10 2 3,2
$20 $1, $20 3 3 3,3
4 3,4
$1 $5, $1
$5 $10 $5, $10 1 4,1
$20 $5, $20 2 4,2
4 3 4,3
4 4,4
$1 $10, $1
$10 $5 $10, $5
$20 $10, $20 34.
Appetizers Entrees Desserts
$1 $20, $1
$20 $5 $20, $5
$10 $20, $10 D1 A1, E1, D1
E1 D2 A1, E1, D2
D3 A1, E1, D3
32. D1 A1, E2, D1
H HHHH
H T HHHT E2 D2 A1, E2, D2
H D3 A1, E2, D3
T H HHT H
T HHTT A1 D1 A1, E3, D1
H E3 D2 A1, E3, D2
H HTHH
H T HTHT D3 A1, E3, D3
T D1 A1, E4, D1
T H HTT H
T HTTT
E4 D2 A1, E4, D2
D3 A1, E4, D3
H THHH D1 A2, E1, D1
H T THHT
H E1 D2 A2, E1, D2
T H THT H D3 A2, E1, D3
T THTT
T D1 A2, E2, D1
H TTHH E2 D2 A2, E2, D2
H T TTHT
T
D3 A2, E2, D3
T H TTT H A2 D1 A2, E3, D1
T TTTT
E3 D2 A2, E3, D2
D3 A2, E3, D3
D1 A2, E4, D1
E4 D2 A2, E4, D2
D3 A2, E4, D3
D1 A3, E1, D1
E1 D2 A3, E1, D2
D3 A3, E1, D3
D1 A3, E2, D1
E2 D2 A3, E2, D2
D3 A3, E2, D3
A3 D1 A3, E3, D1
E3 D2 A3, E3, D2
D3 A3, E3, D3
D1 A3, E4, D1
E4 D2 A3, E4, D2
D3 A3, E4, D3

64
 Chapter 4 - Probability and Counting Rules

35. 38.
English Math Elective Probably

1 1, 1, 1 39.
2 1, 1, 2 The statement is probably not based on
1 3 1, 1, 3 empirical probability and probably not true.
4 1, 1, 4
5 1, 1, 5 40.
The outcomes will be:
1 1, 2, 1 0,0 0,1 0,2 0,3 0,4
2 1, 2, 2 1,0 1,1 1,2 1,3 1,4
1 2 3 1, 2, 3 2,0 2,1 2,2 2,3 2,4
4 1, 2, 4
3,0 3,1 3,2 3,3 3,4
5 1, 2, 5
4,0 4,1 4,2 4,3 4,4
1 1, 3, 1 ' "! # *
a. b. œ c.
2 1, 3, 2 #& #& & #&
3 3 1, 3, 3 "# & "
4 1, 3, 4 d. #&
e. #&
œ &
5 1, 3, 5
41.
1 2, 1, 1 Actual outcomes will vary, however each
2 2, 1, 2 number should occur approximately "' of the
1 3 2, 1, 3 time.
4 2, 1, 4
5 2, 1, 5 42.
Actual outcomes will differ; however, the
1 2, 2, 1 probabilities of 0, 1, and 2 heads will be
2 2, 2, 2 approximately "% ß "# ß and "% respectively.
2 2 3 2, 2, 3
4 2, 2, 4
43.
5 2, 2, 5
a. 1:5, 5:1 e. 1:12, 12:1
1 2, 3, 1
b. 1:1, 1:1 f. 1:3, 3:1
2 2, 3, 2 c. 1:3, 3:1 g. 1:1, 1:1
3 3 2, 3, 3 d. 1:1, 1:1
4 2, 3, 4
5 2, 3, 5 EXERCISE SET 4-2

1.
36.
Two events are mutually exclusive if they
H HH
H T HT
cannot occur at the same time. Examples
will vary.
1 T1
2 T2 2.
T 3 T3 a. No e. No
4 T4 b. No f. Yes
5 T5 c. Yes g. Yes
6 T6 d. No

37. 3.
"ß$%)ß&!$
a. 0.08 a. "ß*!(ß"(# œ !Þ707
b. 0.01
c. 0.08  0.27 œ 0.35 b. %'ß!#%
 "ß!*)ß$("
 #"ß')$
œ
"ß*!(ß"(# "ß*!(ß"(# "ß*!(ß"(#
d. 0.01  0.24  0.11 œ 0.36

65
 Chapter 4 - Probability and Counting Rules

3b. continued 11.

Junior Senior Total
"ß"##ß("#
"ß*!(ß"(#
œ !Þ589 Female 6 6 12
Male 12 4 16
c. #"ß')$
œ !Þ011 Total 18 10 28
"ß*!(ß"(#

") "# ' #% '

d. "ß$*%ß&#(
œ !Þ731 a. #)  #)  #) œ #) œ (
"ß*!(ß"(#
"! "# ' "' %
b. #)
 #)
 #)
œ #)
œ (
4.
#% # #'
$"  $" œ $" ") "! #)
-Þ #)  #) œ #) œ1
5.
% ( "" 12.
"*  "* œ "* Fiction Non-Fiction Total
Adult 30 70 100
6.
$"! "&! %'! #$ Children 100 60 160
*)!
 *)!
œ *)!
or %* 130 130 260
The probability of the event is slightly less "$! "
a. P(fiction) œ œ or 0.5
than 0.5 which makes it about equally likely #'! #
to occur or not to occur. '! $
b. P(children's nonfiction) œ #'! œ "$
7. P(not a children's nonfiction) œ
$
a. $ (
 ## %
 ## œ "% ( "  "$ œ "!
"$ or 0.769
## ## or ""
& ( "# '
b. ##
 ## œ ## or ""
c. $ (
 ## %
 ## œ "% ( c. P(adult book or children's nonfiction) œ
## ## or "" "!! '! "'! )
d. (
 ## œ ## or "#
% "" #'!  #'! œ #'! or "$ or 0.615
##

13.
8.
18 - 24 25 - 34 Total
P(amusement park)  P(water
Male 7922 2534 10,456
park)  P(both) œ 0.95
Female 5779 995 6,774
0.47  0.58  P(both) œ 0.95
1.05  P(both) œ 0.95 Total 13,701 3529 17,230
P(both) œ 0.1 or 10% **&
a. P(female aged 25 - 34) œ "(ß#$!
œ 0.058
9.
P(football or basketball) œ b. P(male or aged 18 - 24) œ
&)  %!  ) *!
#!!
œ #!! or !Þ45
"!ß%&' "$ß(!" (*##
"(ß#$!  "(ß#$!  "(ß#$! œ
*! ""
P(neither) œ "  #!! œ #! or !Þ55
"'ß#$&
"(ß#$! œ !Þ942
10.
a. There are four 4's and 13 diamonds, but c. P(under 25 years and not male) œ
the 4 of diamonds is counted twice; hence,
P(4 or diamond) œ P(4)  P(diamonds)  &((*
œ 0.335
% "(ß#$!
P(4 of diamonds) œ &#  "$ "
&#  &#
"' %
œ &# œ "$ . 14.
Endangered - US Endangered - Foreign
13 13 26 "
b. P(club or diamond) œ 52  52 œ 52 or #
Mammals 68 251
Birds 77 175
4 26 2 Reptiles 14 64
c. P(jack or black) œ 52  52  52 Amphibians 11 8
œ 28 7
52 or 13
Total 170 498

66
 Chapter 4 - Probability and Counting Rules

14. continued 17. continued

Threatened - US Threatened - Foreign Total b. P(is a freshman or is against the issue) œ
Mammals 10 20 349 P(freshman)  P(against)  P(both) œ
Birds 13 6 271 #)
Reptiles 22 16 116 &)
 "#
&)
)
 &) œ $#
&)
or "'
#*
Amphibians 10 1 30
Total 55 43 766 #% "#
c. P(sophomore and in favor) œ &) or #*

a. P(threatened and in the US) œ 18.

&&
(('
œ 0Þ072 1st Class Ad Magazine Total
Home 325 406 203 934
b. P(an endangered foreign bird) œ Business 732 1021 97 1850
"(&
('' œ 0.229 Total 1057 1427 300 2784

c. P(a mammal or a threatened species) œ a. P(home) œ *$%

œ %'(
$%* %$ #! $(# #()% "$*#
(''  (''  ('' œ ('' œ 0.486
b. P(advertisement or business) œ P(ad) 
15. P(business)  P(business and ad) œ
Total œ 136,238 multiple births "%#( ")&! "!#" ##&' %(
#()%  #()%  #()% œ #()% œ &)
a. P(more than two babies) œ
(''$
"$'ß$#) œ 0.056 c. P(1st class or home) œ P(1st class) 
P(home)  P(1st class and home) œ
&&$ "!&( *$% $#& "''' )$$
b. P(quads or quints) œ "$'ß$#)
œ 0.004 #()%  #()%  #()% œ #()% œ "$*#

c. The total number of babies who are 19.

triplets œ 21,330 The total of the frequencies is 30.
# "
The total number of babies from multiple a. $! œ "&
births œ 280,957
#"ß$$! #$& "! "
P(baby is a triplet) œ #)!ß*&( œ 0.076 b. $!
œ $!
œ $

"#)#$ #& &

16. c. $!
œ $!
œ '
Age Male Female Total
"#)#$ #& &
19 and under 4746 4517 9263 d. $!
œ $!
œ '
20 1625 1553 3178
21 1679 1627 3306 )# "! "
e. $!
œ $!
œ $
Total 8050 7697 15,747
20.
a. P(male and 19 or under) œ The total of the frequencies is 32.
%(%' *
"&ß(%(
œ 0.301 a. P(more than 10) œ $# œ 0.281

$!
b. P(20 or female) œ b. P(at least one) œ $#
œ 0.938
$"() ('*( "&&$ *$##
"&ß(%(  "&ß(%(  "&ß(%( œ "&ß(%( œ 0.592
c. P(1 - 5 or more than 15) œ
")
$# œ 0.563
'%)%
c. P(at least 20) œ "&ß(%( œ 0.412

17. 21.
Class Yes No No Opinion Total The total of the frequencies is 24.
Freshmen 15 8 5 28 a. "! &
#% œ "#
Sophomores 24 4 2 30 #" $ "
Total 39 12 7 58 b. #% œ #% œ )

"!$#" "' #
a. P(no opinion) œ ( c. #% œ #% œ $
&)

67
 Chapter 4 - Probability and Counting Rules

21. continued 24.

a. P(sum of 8)  P(sum of 9)  P(sum of
)"!$# #$ & % $
d. #% œ #% 10) œ $'  $'  $' œ "#
$'
or "$

22. b. P(doubles)  P(sum of 7) œ

' ' "# "
$'  $' œ $' or $
High Chol. Normal Chol. Total
Alcoholic 87 13 100
Non-Alcoholic 43 157 200 c. P(sum > 9)  P(sum < 4) œ
' $ * "
Total 56 244 300 $'  $' œ $' or %

a. P(alcoholic with elevated d. The event in part c is least likely to occur

)( #*
cholesterol) œ $!! œ "!! since it has the lowest probability.

#!! # 25.
b. P(non-alcoholic) œ $!! œ $
P(export or
c. P(non-alcoholic with normal ethanol) œ "Þ*
"" 
"Þ'
""  !
"" œ 0.318
cholesterol) œ "&(
$!!
26.
23. There are 6$ œ 216 possible outcomes.
' "
a. There are 4 kings, 4 queens, and 4 jacks; a. #"' œ $' since there are 6 triples: (1,1,1),
hence P(king or queen or jack) œ "#
&# œ "$
$ (2,2,2), . . . , (6,6,6).
' "
b. There are 13 clubs, 13 hearts, and 13 b. #"' œ $' since there are six possible
spades; hence, P(club or heart or spade) œ outcomes summing to 5: (1,2,2), (2,1,2),
"$"$"$ $*
&#
œ &# œ %$ (2,2,1), (1,1,3), (1,3,1), and (3,1,1).

c. There are 4 kings, 4 queens, and 13 27.

diamonds but the king and queen of P(mushrooms or pepperoni) œ
diamonds were counted twice. P(mushrooms)  P(pepperoni) 
Hence; P(king or queen or P(mushrooms and pepperoni)
diamond) œ P(king) 
P(queen)  P(diamond)  P(king and queen Let X = P(mushrooms and pepperoni)
of diamonds) œ &#%
 &#%
 "$ # "* Then 0.55 œ 0.32  0.17  X
&#  &# œ &#
X œ 0.06
d. There are 4 aces, 13 diamonds, and 13
hearts. There is one ace of diamonds and 28.
one ace of hearts. P(one or two car garage) œ
Hence, P(ace or diamond or 0.20  0.70 œ 0.90
heart) œ P(ace)  P(diamond)  P(heart) Hence, P(no garage) œ 1  0.90 œ 0.10
 P(ace of hearts and ace of diamonds) œ
% "$ "$ # #) ( 29.
&#  &#  &#  &# œ &# œ "$
P(not a two-car garage) œ 1  0.70 œ 0.30
e. There are 4 nines, 4 tens, 13 spades, and
30.
13 clubs. There is one nine of spades, one
No. P(A  B) Á 0
ten of spades, one nine of clubs and one ten
of clubs.
EXERCISE SET 4-3
Hence, P(9 or 10 or spade or
club) œ P(9)  P(10)  P(spade)  P(club)
1.
 P(9 and 10 of clubs and spades) œ
% % "$ "$ % $! "& a. Independent e. Independent
&#  &#  &#  &#  &# œ &# œ #' b. Dependent f. Dependent
c. Dependent g. Dependent
d. Dependent h. Independent

68
 Chapter 4 - Probability and Counting Rules

2. 11. continued
P(all 5 exercise b. P(all three had cable
regularly) œ (0.37)& œ 0.007 or 0.7% TV) œ (0.86)$ œ 0.636
The event is unlikely to occur since its
probability is very small. c. P(at least one had cable TV)
œ 1  0.003
3. œ 0.997
a. P(none play video or computer
games) œ (0.31)% œ 0.009 or 0.9% 12.
# " "
P(both are defective) œ ' † & œ "&
b. P(all four play video or computer
games) œ (0.69)% œ 0.227 or 22.7% 13.
% $ # " "
a. &# † &" † &! † %* œ #(!ß(#&
4.
P(all 4 used a seat belt) œ (.52)% œ 7.3% b. "$
† "#
† ""
† "!
œ ""
&# &" &! %* %"'&

5. #' #& #% #$ %'

c. &# † &" † &! † %* œ )$$
P(making a sale) œ 0.21
P(making 4 sales) œ (0.21)%
œ 0.00194 ¸ 0.002 14.
& % $ &
) † ( †
The event is unlikely to occur since the '
œ #)
probability is small.
15.
$ # " "
) † ( †
6. '
œ &'
P(two inmates are not citizens) œ (0.25)#
œ 0.0625 or 6.3% 16.
P(both prizes are won by men) œ
"! * *! $
$! † #* œ )(! or #*
7. unlikely
P(all are citizens) œ (0.801)$ œ 0.514
17.
8. P(both prizes are won by women) œ
#! "*
a. P(at least one doesn't use a computer at † œ $)
$! #* )(
more likely than in #16.
work) œ 1  P(none of the women don't use
a computer at work) 18.
P(at least one doesn't use a D (0.8)(0.1) = 0.08
computer) œ 1  (0.72)& œ 0.807 0.1
0.8 Model I
b. P(all 5 use a 0.9
computer) œ (0.72)& œ 0.194
ND
D (0.2)(0.18) = 0.036
9.
P(none use phones for 0.18
texting) œ (0.65)% œ 0.179 0.2 Model II
0.82
10. ND
"$ "# "
a. P(both are spades) œ &# † &" œ "(
P(defective) œ 0.08  0.036 œ 0.116
b. P(both are the same suit) œ
% "# %
% † &" œ "(

% $ "
c. P(both are kings) œ &#
† &"
œ ##"

11.
a. P(none had cable TV) œ (0.14)$ œ 0.003

69
 Chapter 4 - Probability and Counting Rules

19. 22. continued

Fed. Aid (0.448) P(defective | classified
Aid (0.606) good) œ Ð!Þ!"&!Þ!"&
 !Þ('&Ñ œ 0.019
Male (0.4237) No Fed. Aid (0.552)
No Aid (0.394) 23.
Fed. Aid (0.504) !Þ!(
P(female | adult) œ !Þ** œ 0.0.071
Aid (0.652)
Female (0.5763) No Fed. Aid (0.496) 24.
No Aid (0.348) P( Ÿ 9 1st roll and Ÿ 9 2nd roll and >9 3rd
roll) œ $! † $! † ' œ 0.116
$' $' $'
a. P(male student without
aid) œ 0.4237(0.394) œ 0.0167 25.
P(2nd defective | 1st defective) œ
b. P(male student | student has aid) œ
# "
P(1st and 2nd defective) )†( "
P(1st defective)
œ # œ (
P(aid & male) !Þ%#$(Ð!Þ'!'Ñ
P(aid) œ !Þ%#$(Ð!Þ'!'Ñ!Þ&('$Ð!Þ'&#Ñ œ 0.406 )

26.
c. P(female student or a student who P(swim | bridge) œ P(play bridge and swim)
P(play bridge)
receives federal aid) œ
P(female)  P(federal aid)  P(female with !Þ($
federal aid) œ œ !Þ)#
œ 0.89 or 89%
0.5763  (0.115  0.1894)  0.1894 œ 0.69
27.
!Þ!%#
20. P(calculus | dean's list) œ !Þ#"
œ 0.2
" 5 " $ " % %*
P(red) œ $
† 8
 $
† %
 $
† '
œ (#
28.
!Þ(#
21. P(graduate | play golf) œ !Þ)!
œ 0.9
Risk
A (0.6)(0.01) = 0.006 29.
!Þ'&
0.01 P(salad | pizza) œ !Þ*&
œ 0.684 or 68.4%
Low
0.6 0.99 30.
NA a. P(coffee or
A (0.3)(0.05) = 0.015 candy) œ %$
(( 
##
((
 "!
((
œ 0.714
0.05
0.3 Medium "!Î((
b. P(tea | contains mugs) œ #$Î((
œ 0.435
0.95
NA "#
c. P(tea and cookies) œ (( œ 0.156
A (0.1)(0.09) = 0.009
0.1 0.09
31.
High
a. P(O ) œ 0.06
0.91
b. P(type O l Rh ) œ !Þ$(
!Þ)&
œ 0.435
NA
c. P(A or AB ) œ 0.34  0.01 œ 0.35
d. P(Rh | type B) œ !Þ!#
!Þ"# œ 0.167
P(accident) œ .006  .015  .009 œ 0.03
32.
22. $$ß!#!
"#%ß'%&
P(defective) œ 0.15 a. P(male | pediatrician) œ ''ß$(" œ 0.498
"#%ß'%&
P(defective &
misclassified) œ (0.15)(0.1) œ 0.015 b. P(pathologist | female) œ &'!%
œ 0.109
&"ß#%(
P(good & correctly classified) œ
(0.85)(0.9) œ 0.765
c. No. P(pathologist | female) Á P(female)
P(good) œ 0.765  0.015 œ 0.78

70
 Chapter 4 - Probability and Counting Rules

33. 39. continued

$'
a. P(gold | U. S.) œ *&(
""! œ 0.327 œ 1  P(all 5 on time)
*&( œ 1  (0.843)& œ 0.574
36
b. P(U. S. | gold) œ 957
302 œ 0.119 40.
957
P(male) œ 1  0.64 œ 0.36
c. No. P(gold | U. S.) Á P(gold) P(online gamer and
male) œ (0.56)(0.36) œ 0.202
34.
a. P(no computer) œ 1  0.543 œ 0.457 41.
P(none of three has a computer) œ If P(read to) œ 0.58, then
(0.457)$ œ 0.095 P(not being read to) œ 1  0.58 œ 0.42

b. P(at least one has a computer) œ P(at least one is read to) œ 1  P(none are
1  P(none of three has a computer) œ read to)
1  0.0954 œ 0.905 œ 1  P(all five are not read to)
œ 1  (0.42)& œ 0.987
c. P(all three have computers) œ
(0.543)$ œ 0.160 42.
a. P(all three have
35. assistantships) œ (0.6)$ œ 0.216
a. P(all 3 get enough
exercise) œ (0.27)$ œ 0.020 b. P (none have
assistantships) œ (0.4)$ œ 0.064
b. P(at least one gets enough
exercise) œ 1  (0.73)$ œ 0.611 c. P(at least one has an
assistantship) œ 1  (none have
36. assistantships)
P(5 buy at least 1) œ œ 1  0.064 œ 0.936
*! )* )) )( )'
"#! † ""* † "") † ""( † ""' œ 0.231
43.
37. P(at least one club) œ 1  P(no clubs)
a. P(none have been "  $* $) $( $' '$#(
&# † &" † &! † %* œ "  #!ß)#&
"%ß%*)
married) œ (0.703)& œ !Þ"(2 œ #!ß)#&

b. P(at least one has been married) œ 44.

1  P(none have been married) P(part-time) œ 1  0.81 œ 0.19
œ 1  0.1717 P(student at a four-year college and part-
œ 0.828 time) œ (0.69)(0.19) œ 0.131

38. 45.
a. P(all three caused by driver a. P(not a family and children's
error) œ (0.54)$ œ 0.157 game) œ 1  0.198 œ 0.802
P(none of five are family and children's
b. P(none caused by driver games) œ (0.802)& œ 0.332
error) œ (0.46)$ œ 0.097
b. P(at least one is family and children's
c. P(at least one caused by driver game) œ 1  0.332 œ 0.668
error) œ 1  P(none by driver error)
œ 1  0.0973 œ 0.9027 or 0.903 46.
P(at least one will not improve) œ " 
39. P(all will improve) œ "  Ð!Þ(&Ñ"#
P(at least one not on time) œ œ 0.968 or 96.8%
1  P(none not on time)

71
 Chapter 4 - Probability and Counting Rules

47. 55.
P(at least one tail) œ 1  P(no tails) Yes.
"  Ð "# Ñ& œ "  $#
"
œ $"
$#
P(enroll) œ 0.55
48.
P(at least one X) œ 1  P(no X's) P(enroll | DW)  P(enroll) which indicates
"  Ð #&
#'
Ñ$ œ "  "&ß'#&
"(ß&('
"*&"
œ "(ß&(' or 0.111 that DW has a positive effect on enrollment.
The event is unlikely to occur since the
probability is only about 11%. P(enroll | LP) œ P(enroll) which indicates
that LP has no effect on enrollment.
49.
P(rolling a 4) œ "' P(enroll | MH)  P(enroll) which indicates
that MH has a low effect on enrollment.
P(at least one 4) œ 1  P(no fours)
P(at least one 4) œ 1  P(all 6 are not 4's)
Thus, all students should meet with DW.
P(at least one 4) œ 1  ( &' )' œ 0.665
It will happen almost 67% ( #$ ) of the time. 56.
It's somewhat likely. P(buy) œ 0.35

50. a. If P(buy | ad) œ 0.20, then the

a. P(all 6 had A commercial adversely effects the probability
averages) œ (0.47)' œ 0.011 of buying since the events are dependent and
the probability that a person buys the
b. P(none had A product is less than 0.35. The events are
averages) œ (0.53)' œ 0.022 dependent.

c. P(at least one had an A b. If P(buy | ad) œ 0.35, then the

average) œ 1  (0.53)' œ 0.978 commercial has no effect on buying the
product. The events are independent.
51.
P(at least one even) œ 1  P(no evens) c. If P(buy | ad) œ 0.55, then the
"  Ð "# Ñ$ œ "  ") œ () commercial has an effect on buying the
product. The events are dependent.
52.
P(at least one rose) œ 1  P(no roses) EXERCISE SET 4-4
"  #' #& #% #$ (%(&
$% † $$ † $# † $" œ "  #$ß")) œ 0.678
1.
53. 10& œ 100,000
No, because P(A  B) œ 0 and 10 † 9 † 8 † 7 † 6 œ 30,240
P(A  B) Á P(A) † P(B)
2.
54. 9! œ 9 † 8 † 7 † 6 † 5 † 4 † 3 † 2 † 1 œ 362,880
If independent, then P(compact |
domestic) œ P(compact) 3.
6! œ 6 † 5 † 4 † 3 † 2 † 1 œ 720
P(compact) œ "&! "
$!! œ #
P(compact | domestic) œ P(domestic and compact) 4.
P(domestic)
"!! 9! œ 9 † 8 † 7 † 6 † 5 † 4 † 3 † 2 † 1 œ 362,880
"!! "!
œ $!!
#"! œ #"!
or #"
$!!
5.
Thus, P(compact | domestic) Á P(compact) 7! œ 7 † 6 † 5 † 4 † 3 † 2 † 1 œ 5040
since "# Á "!
#" .
6.
5! œ 120
3 † 2 † 2 † 1 † 1 † 1 œ 12

72
 Chapter 4 - Probability and Counting Rules

7. 13. continued
10! œ 3,628,000 j. ' P# œ Ð' 'x#Ñx

8. œ '†&†%†$†#†"
œ 30
%†$†#†"
2 † 25 † 24 † 23 œ 27,600
2 † 26 † 26 † 26 œ 35,152 14.
)x )x
) P) œ Ð))Ñx
œ !x
œ 40,320
9.
10 † 10 † 10 œ 1000
1 † 9 † 8 œ 72 15.
%x %†$†#†"
% P% œ Ð%%Ñx
œ !x
œ 24
10.
If repetitions are permitted: 6% œ 1296 16.
(x (†'†&†%x
If repetitions are not permitted: ( P$ œ Ð($Ñx œ %x œ 210
6 † 5 † 4 † 3 œ 360
17.
11. ##x
## C% œ Ð##%Ñx %x
œ 7315
6 † 5 † 5 † 4 œ 600
18.
12. "!x "!x
2†4 œ 8 "! P& œ Ð"!&Ñx
œ &x
œ 30,240

13. 19.
(x (†'†&†%†$†#†"
a. 8x œ 8 † 7 † 6 † 5 † 4 † 3 † 2 † 1 œ 40,320 ( P% œ Ð(%Ñx
œ $†#†"
œ 840

b. 10x œ 10 † 9 † 8 † 7 † 6 † 5 † 4 † 3 † 2 † 1 20.
10x œ 3,628,800 4 † 6 † 5 œ 120

c. 0x œ 1 21.
"!x "!†*†)†(†'†&†%†$†#†"
"! P' œ Ð"!'Ñx
œ %†$†#†"
œ 151,200
d. 1x œ 1

(x
22.
e. ( P& œ Exactly 3 samples: œ "$x
œ 286
Ð(  &Ñx "$ C$ Ð"$  $Ñx $x

(†'†&†%†$†#†"
œ #†"
œ 2520 Up to 3 samples:
"$ C!  "$ C"  "$ C#  "$ C$ œ 378
"#x
f. "# P% œ Ð"#  %Ñx
23.
"#†""†"!†*†)†(†'†&†%†$†#†" &!x &!x
œ )†(†'†&†%†$†#†"
œ 11,880 &! P% œ Ð&!%Ñx œ %'x œ 5,527,200

&x
g. & P$ œ Ð&  $Ñx 24.
""x
"" C% œ Ð""%Ñx %x
œ 330
&†%†$†#†"
œ #†" œ 60
25.
'x "#x
h. ' P! œ Ð'  !Ñx Same task: "# C% œ Ð"#  %Ñx %x œ 495

'†&†%†$†#†" "#x
œ '†&†%†$†#†" œ1 Different tasks: "# P% œ Ð"#  %Ñx œ 11,880

&x
i. & P& œ Ð&  &Ñx 26.
(x (x (†'†&†%†$†#x
( P& œ Ð(&Ñx
œ #x
œ #x
œ 2520
&†%†$†#†"
œ !x œ 120

73
 Chapter 4 - Probability and Counting Rules

27. 37. continued

a. $x&x#x œ 10 f. $x
$x !x œ1 ( C# † & C#  ( C$ † & C"  ( C% œ
21 † 10  35 † 5  35 œ
b. )x
œ 56 g. $x
œ1 210  175  35 œ 420
&x $x !x $x

c. (x
œ 35 h. *x
œ 36 38.
$x %x #x (x "!x "!x
"! C$ † "! C$ œ (x $x † (x $x
"!†*†)†(x "!†*†)†(x
d. 'x
œ 15 i. "#x
œ 66 œ (x † $†#†" † (x † $†#†" œ 120 † 120 œ 14,400
4x 2x "!x #x

e. 'x
œ 15 j. %x
œ4 39.
#x %x "x $x
The possibilities are CVV or VCV or VVV.
28.
&#x &#†&"†&!†%*x Assuming the same vowel can't be used
&# C$ œ %*x $x œ %*x † $†#†" œ 22,100
twice in a "word":
7 † 5 † 4  5 † 7 † 4  5 † 4 † 3 œ 340
29.
6x '†&†%†$x
' P$ œ $x œ $x œ 120 Assuming the same vowel can be used twice
in a "word":
30. 7 † 5 † 5  5 † 7 † 5  5 † 5 † 5 œ 475
"#x *x
"# C% † * C$ œ )x %x
† 'x $x
40.
"#†""†"!†*†)x *†)†(†'x "#x "!x
œ )x † %†$†#†"
† 'x †$†#†"
œ 41,580 "# C' † "! C' œ 'x 'x
† %x 'x

31. "#†""†"!†*†)†(†'x "!†*†)†(†'x

œ 'x † '†&†%†$†#†"
† 'x † %†$†#†"
"!x
"! C% œ 'x %x
œ 210
œ 924 † 210 œ 194,040
32.
"!x "!†*†)†(x
"! C$ œ (x $x
œ (x $†#†"
œ 120 41.
"!x "#x
"! C# † "# C# œ )x #x † "!x #x
33. œ 45 † 66 œ 2,970
#!x
#! C& œ "&x &x
œ 15,504
42.
#&x #&†#%†#$†##†#"†#!x
34. #& C& œ #!x &x œ #!x † &†%†$†#†"
""x ""†"!†*†)†(†'x
"" C' œ &x 'x
œ 'x †&†%†$†#†"
œ 462 œ 53,130

35. 43.
Different programs: There are ( C# œ 21 tiles with unequal
")x numbers and 7 tiles with equal numbers.
") C"! œ Ð")  "!Ñx "!x œ 43,758
Thus, the total number of tiles is 28.
Starting and ending with the same song:
"'x 44.
"' C) œ Ð"'  )Ñx )x œ 12,870 "'x "&x
"' C% † "& C# œ Ð"'  %Ñx %x
† Ð"&  #Ñx #x

36.
%x "#x (x œ 191,100
% C# † "# C& † ( C$ œ #x #x † (x &x † %x $x

%†$†#x "#†""†"!†*†)†(x (†'†&†%x 45.

œ #x † #†" † (x † &†%†$†#†" † %x † $†#†" ""x ""†"!†*†)†(x
"" C( œ %x (x œ (x † %†$†#†" œ 330
œ 6 † 792 † 35 œ 166,320
46.
"$x "$†"#†""†"!†*†)x
"$ C) œ œ œ 1287
37. &x )x )x†&†%†$†#†"

"# C% œ 495
47.
( C# † & C# œ 21 † 10 œ 210 "!x "#x
"! C' † "# C' œ %x 'x † 'x 'x œ 194,040

74
 Chapter 4 - Probability and Counting Rules

48. 62. continued

œ *x"(x)x œ
"( C)
"(†"'†"&†"%†"$†"#†""†"!†*x
*x † )†(†'†&†%†$†#†"
B C 3 2 1 = 6
œ 24,310 B 3 C 2 1 = 6
B 3 2 C 1 = 6
49. B 3 2 1 C = 6
#!x 3 B C 2 1 = 6
#! C) œ Ð#!  )Ñx )x
œ #!†"*†")†"(†"'†"&†"%†"$†"#x
œ 125,970 3 B 2 C 1 = 6
"#x †)†(†'†&†%†$†#†"
3 B 2 1 C = 6
50. 3 2 B C 1 = 6
'x '†&†%†$†#†" 3 2 B 1 C = 6
' P$ œ $x
œ $†#†"
œ 120
3 2 1 B C = 6
60
51.
#!x #!†"*†")†"(†"'†"&x
#! P& œ "&x
œ "&x
œ 1,860,480
c. 5x  2 † 4x œ 72
52.
""x )x 63.
"" C# † ) C$ œ † œ 3080
Ð""  #Ñx #x Ð)  $Ñx $x a. % C" † 1 † 1 † 1 † 1 œ 4
b. "! C" † % C"  % C" œ 36
53. c. "$ C" † "# C" † % C" œ 624
"(x "(†"'†"&x
"( C# œ Ð"(  #Ñx #x
œ "&x †#†"
œ 136 d. "$ C" † % C$ † "# C" † % C# œ 3744

54.
"!x "!†*†)x EXERCISE SET 4-5
"! C) œ Ð"!  )Ñx )x
œ #†"†)x
œ 45

55. 1.
"# "" ""
&x &†%†$†#†" P(2 face cards) œ † œ
& P& œ !x
œ "
œ 120 &# &" ##"

56. 2.
&†%x
&x &x &x & C% "
& P$  & P%  & P& œ   œ 300 a. #& C%
œ %x†"
#&†#%†#$†##†#"x œ #&$!
#x "x !x #"x†%†$†#†"

57. & C# †#! C#

&†%†$x #!†"*†")x
$x†#†" † ")x†#†" $)
'x &x b. #& C%
œ #&†#$†##
œ #&$
' C$ † & C# œ $x $x
† $x #x
œ 200
#!†"*†")†"(†"'x
#! C% *'*
58. c. #& C%
œ "'x†%†$†#†"
#&†#$†##
œ #&$!
*x *†)†(†'†&x
* C& œ %x &x
œ %†$†#†"†&x
œ 126
&†%x #!†"*†")†"(x
& C" †#! C$ %x†" † "(x†$†#†" ""%
d. #& C%
œ #&†#$†##
œ #&$
59.
)x )†(†'†&†%†$†#†"
) P$ œ &x
œ &†%†$†#†"
œ 336 3.
a. There are % C$ ways of selecting 3 women
60. and ( C$ total ways to select 3 people;
% C"  % C#  % C$  % C% œ 15 hence, P(all women) œ %( CC$$ œ $&
%
Þ

61. b. There are 3 C$ ways of selecting 3 men;

1†2†1 œ 2 hence, P(all men) œ $( CC$$ œ $&
"
Þ
1†3†2†1 œ 6
1 † (8  ") † (8  #) † † 3 † 2 † 1 œ (8  ")!
c. There are $ C# ways of selecting 2 men
and % C" ways of selecting one woman;
62.
a. 2! † 4! œ 48 hence, P(2 men and 1 woman) œ $ C(#C†%$C"
b. 60 ways œ "#
$& Þ
Using a table, list the number of ways in
each column and multiply:

75
 Chapter 4 - Probability and Counting Rules

3. continued 9.
d. There are $ C" ways to select one man and a. P(one of each) œ
") C" †"! C"† $ C" &%!
% C# ways of selecting two women; hence, œ %%*& œ 0.120
$" C$
P( 1 man and 2 women) œ $ C("C†%$C# œ ")
$&
Þ
b. P(no Navy members) œ
#" C$
4. There are %" C$ ways to select 3
$" C$
œ "$$!
%%*&
œ 0.296
Republicans; hence, P(3 Republicans) œ
%" C$ "!ß''!
"!! C$
œ "'"ß(!! œ !Þ066 c. P(three Marines) œ
") C$ †"! C! †$ C! )"'
$" C$
œ %%*& œ 0.182
There are &( C$ ways to select 3 Democrats;
&( C$
hence P(3 Democrats) œ "!! C$ 10.
#*ß#'!
P(3 Democrats) œ "'"ß(!! œ !Þ181 There are 6 red face cards and 16 black
cards numbered 2 - 9, for a total of 22 cards.
There are %" C" ways to select one
' C% †"' C!
Republican, # C" ways to select one a. P(all 4 red) œ ## C%
œ 0.002
Independent, and &( C" ways to select one
' C# †"' C#
Democrat; hence P(one from each b. P(2 red and 2 black) œ ## C%
œ 0.246
†# C" †&( C"
party) œ %" C""!! C$
%'(%
œ "'"ß(!! œ !Þ029
c. P(at least one red) œ 1  P(none red)
5. P(at least one red) œ 1  ' C##! †C"'%C% œ 0.751
a. P(all Republicans) œ
&" C$ † %) C! † " C! #!ß)#& "' C% †' C!
"!! C$
œ "'"ß(!! œ 0.129 d. P(all 4 black) œ œ 0.249
## C%

b. P(all Democrats) œ 11.

&" C! †%) C$ †" C! "(ß#*' "" C# &&
"!! C$
œ "'"ß(!! œ 0.107 a. P(red) œ œ "("
œ 0.322
"* C#

c. P(one Republican, Democrat, and ) C# #)

b. P(black) œ "* C#
œ "("
œ 0.164
Independent) œ
&" C" †%) C" †" C" #%%)
"!! C$
œ "'"ß(!! œ 0.015 c. P(unmatched) œ "" C" †) C"
œ ))
œ 0.515
"* C# "("

6. d. It probably got lost in the wash!

a.P(no defective
resistors) œ "#* CC%% œ "#'
%*&
œ "%
&& 12.
) C$ †* C% &'†"#' (!&' ))#
"( C(
œ "*ß%%)
œ "*ß%%)
œ #%$"
b.P(1 defective
resistor) œ $ C"#"C†*%C$ œ #&#
%*&
œ #)
&& 13.
There are 6$ œ 216 ways of tossing three
$ C$ †* C" "
c.P(3 defective resistors) œ "# C%
œ && dice, and there are 15 ways of getting a sum
of 7; i.e., (1, 1, 5), (1, 5, 1), (5, 1, 1),
7. (1, 2, 4), etc. Hence the probability of
"& &
#
† "
œ " rolling a sum of 7 is #"' œ (# Þ
&! %* "##&

8. 14.
There are % C$ ways of getting 3 of a kind for a. P(all 4 seniors)
one denomination and there are 13 œ "! C% †#! C'&!C†#!%C! †"& C! œ 0.0003
denominations. There are % C# ways of
getting two of a kind and 12 denominations b. P(one of each)
left. There are &# C& ways to get five cards; œ #! C" †#! C'&"C†"&%C" †"! C" œ 0.089
hence,
P(full house) œ "$†% C&#$C†"#†
&
% C# 6
œ 4165 c. P(2 sophomores and 2 freshmen)
œ #! C# †#! C'&#C†"&%C! †"! C! œ 0.053

76
 Chapter 4 - Probability and Counting Rules

14. continued 6.
*
d. P(at least 1 senior) a. $&
œ 1  P(none are seniors)
C% ( "' #$
œ 1  &&
'& C%
b. $&  $& œ $&
œ 0.496
$ ( * "*
c. $&
 $&
 $&
œ $&
15.
There are 5! œ 120 ways to arrange 5 "' "*
d. "  $& œ $&
washers in a row and 2 ways to have them in
correct order, small to large or large to 7.
# "
small; hence, the probability is "#! œ '! Þ P(either air-conditioning or CD player)
œ 0.5  0.37  0.06 œ 0.81
16. P(neither air-conditioning nor CD)
&#x
There are &# C& œ %(x &x
œ 2,598,960 possible œ 1  0.81 œ 0.19
hands.
% $' 8.
a. #ß&*)ß*'! b. #ß&*)ß*'!
Refer to the sample space for tossing two
c. '#% dice.
#ß&*)ß*'!

a. There are 4 ways to roll a 5 and 5 ways to

17. % &
roll a 6; hence, P(5 or 6) œ $'  $' œ "%
P(berries are produced) œ P(either 1 or 2
males)
b. There are 3 ways to get a 10, 2 ways to
P(1 or 2 males) œ ) C"##C†%$C"  ) C"#"C†%$C#
get an 11 and 1 way to get a 12; hence,
œ 0.509  0.218 œ 0.727 $
P(sum greater than 9) œ $' #
 #' "
 $' œ "'
REVIEW EXERCISES - CHAPTER 4
c. A sum less than 4 means 3 or 2, and
greater than 9 means 10, 11, 12; hence, the
1.
" # $ " probability is #"$#"
$'
*
œ $' œ %" Þ
a. '
b. $
c. '
œ #

d. Four, 8, and 12 are divisible by 4; hence,

2.
"$ " % " the probability of rolling a 4, 8, or 12 is
a. œ d. œ $&" *
&# % &# "$
$'
œ $' œ %" Þ
"" #' "
b. #'
e. &#
œ # e. Since this is impossible, the answer is 0.
"
c. &# f. Since this is the entire sample space, the
probability is $'
$' œ "Þ
3.
a. P(not used for taxes) œ P(virus or other) 9.
& #
P(virus or other) œ "!  "! œ 0.7 0.80 + 0.30  0.12 œ 0.98
$ #
b. P(taxes or other use) œ "!  "! œ 0.5 10.
P(John or Mary) œ
4. P(John) + P(Mary)  P(John and Mary) œ
#
P(even) œ ' œ 0.333 0.39 + 0.73  0.36 œ 0.76
The probability that neither purchases a new
"'
P(odd sum) œ $' œ 0.444 car is 1  0.76 œ 0.24

5. 11.
"$ "
P(preferred juice) œ '!
P(enrolled in an online course) œ '
or 0.167
a. P(all 5 took an online
course) œ ( "' )& œ 0.0001

77
 Chapter 4 - Probability and Counting Rules

11. continued 17.

b. P(none took an online D (0.25)(0.1) = 0.025
course) œ ( &' Ñ& œ 0.402 0.1
V
0.25 0.9
c. P(at least one took an online course) ND
œ 1  P(none took an online course) D (0.75)(0.5) = 0.375
œ 1  ( &' Ñ& œ 0.598 0.75 0.5
NV
0.5
12.
ND
P(five borrowed books) œ (0.67)& œ 0.135

P(none borrowed books) œ (0.33)& œ 0.004 P(disease) œ 0.025  0.375 œ 0.4

13. 18.
Model
a. #'
&#
† #&
&"
† #%
&!
œ #
"( S (0.4)(0.03) = 0.012
0.03
"$ "# "" $$ ""
b. &#
† &"
† &!
œ #&&!
œ )&!
A
0.97
% $ # " NS
c. &#
† &"
† &!
œ &&#& 0.4 S (0.4)(0.07) = 0.028
0.07
14. 0.4 B
a. "# † %
&#
œ "
#'
0.93
0.2 NS
" #' " S (0.2)(0.09) = 0.018
b. #
† &#
œ % 0.09
C
" "$ " 0.91
c. #
† &#
œ )
NS

15.
Total number of movie releases œ 1384 P(stereo) œ 0.012  0.028  0.018 œ 0.058
or 5.8%
)$%
a. P(European) œ "$)%
œ 0.603
19.
P(NC and C) !Þ$(
b. P(US) œ %("
œ 0.340 P(NC | C) œ P(C)
œ !Þ($
œ 0.507
"$)%

$"' "$# 20.

c. P(German or French) œ "$)%
 "$)% "
%%)
œ "$)% or 0.324 P(all 4 correct) œ #%
œ 0.042

d. P(German | European) P(3 are correct) œ 0, since if 3 labels are

P(European and German)
$"' correct, the 4th label must also be correct.
œ P(European) œ "$)%
)$% œ 0.379
"$)%
'
P(2 are correct) œ #% œ 0.25
16.
X Y Z Total P(at least one correct) œ 1  P(none
TV 18 32 15 65 correct)
Stereo 6 20 13 39 P(at least one correct) œ 1  P(all 4 labels
Total 24 52 28 104 are wrong)
*
P(at least one correct) œ 1  #% œ "&
#%
or
#% $* ' &(
a. "!%  "!%  "!% œ "!% 0.625
&# #) )! "!
b. "!%  "!% œ "!% œ "$ 21.
!Þ%$
!Þ(&
œ 0.573 or 57.3%
'& #) "& () $
c. "!%  "!%  "!% œ "!% œ %

78
 Chapter 4 - Probability and Counting Rules

22. 30.
P(bus late | bad weather) œ )x )x
8! œ ) P) œ Ð)  )Ñx œ !x œ 40,320
P(bus late and bad weather)
P(bad weather) œ !Þ!#$
!Þ%! œ 0.058
31.
"!x
23. "! C# œ )x #x œ 45
<4 yrs HS HS College Total
Smoker 6 14 19 39 32.
18 7 25 50 'x &x %x
Non-Smoker ' C$ † & C# † % C" œ $x $x
† $x #x
† $x "x
Total 24 21 44 89
œ 20 † 10 † 4 œ 800
a. There are 44 college graduates and 19 of
them smoke; hence, the probability is "*
%% Þ 33.
100x
b. There are 24 people who did not graduate
from high school, 6 of whom do not smoke; 34.
hence, the probability is 5 † 3 † 2 œ 30
' "
#% œ % Þ
35.
"#x "#†""†"!†*†)x
24. "# C% œ )x %x
œ %†$†#†"†)x
œ 495
P(veteran) œ 0.11; P(not a veteran) œ 0.89
P(none of 5 are veterans) œ (0.89)& œ 0.558 36.
P(at least one is a "$x "$†"#†""†"!x
"$ C$ œ "!x $x
œ "!x † $†#†"
œ 286
veteran) œ 1  0.558 œ 0.442
37.
25. #! C& œ
#!x
œ #!†"*†")†"(†"'†"&x
œ 15,504
"&x &x "&x &†%†$†#†"
P(at least one household has no DVD
player) 38.
œ 1  P(none have no DVD player) 3 † 5 † 4 œ 60
œ 1  P(all 6 have DVD players)
œ 1  (0.81)' œ 0.718 39.
Total number of outcomes:
26. 26 † 26 † 26 † 10 † 10 † 10 † 10 œ 175,760,000
P(at least one has chronic sinusitis) œ Total number of ways for USA followed by
"  P(none has chronic sinusitis) a number divisible by 5:
"  Ð!Þ)&Ñ& œ 0.556 or 55.6% 1 † 1 † 1 † 10 † 10 † 10 † 2 œ 2000
2000
Hence P œ 175,760,000 œ 0.00001
27.
If repetitions are allowed:
40.
#' † #' † #' † "! † "! † "! œ 175,760,000
There are $ C# ways of attending two plays
and & C" ways of attending one movie, and a
If repetitions are not allowed:
#'†#&†#%†#$x "!†*†)†(†'x total of "! C$ of attending 3 events; hence, the
#' P$ † "! P% œ †
#$x 'x probability is:
œ 78,624,000 $ C# †& C" "&
œ "#! œ ")
"! C$

If repetitions are allowed in the letters but

not in the digits: 41.
26 † 26 † 26 † "! P% œ 88,583,040 Total number of territories œ 45
P(3 French or 3 UK or 3
C$ C$ C$
28. US) œ "'
%& C$
 "&
%& C$
 "%
%& C$

5 † 11 † 2 † 2 œ 220 different types of paper

&'! %&& $'%
œ "%ß"*!  "%ß"*!  "%ß"*!
29.
&x (x "$(*
& C$ † ( C% œ #x $x
† $x %x
œ 10 † 35 œ 350 œ "%ß"*! œ 0.097

79
 Chapter 4 - Probability and Counting Rules

42. 16. b
P(Yahtzee on first roll) œ 17. b
' " " " "
' † ' † ' † ' † ' œ 0.000772 or 0.0008
18. Sample space
19. 0, 1
P(Yahtzee on two successive rolls) œ 20. 0
(0.000772)# œ 0.0000006 21. 1
22. Mutually exclusive
43.
% " "' %
A M, S, A 23. a. &#
œ "$
c. &#
œ "$
S Fa M, S, Fa
St M, S, St
% "
b. &#
œ "$
A M, Ma, A
Ma Fa M, Ma, Fa
"$ "
St M, Ma, St 24. a. &#
œ %
d. 4
="
&# "$
M
A M, D, A %"$" % #' "
D Fa M, D, Fa b. &#
œ "$
e. &#
œ #
St M, D, St
"
A M, W, A c. &#
W Fa M, W, Fa
St M, W, St "# #(
25Þ a. $"
c. $"
A F, S, A
S Fa F, S, Fa "# #%
St F, S, St
b. $"
d. $"

A F, Ma, A "" "

Ma Fa F, Ma, Fa
26. a. $'
d. $
St F, Ma, St
F &
b. ")
e. 0
A F, D, A
D Fa F, D, Fa
"" ""
St F, D, St c. $'
f. "#
A F, W, A
W Fa F, W, Fa 27. (0.75  0.16)  (0.25  0.16) œ 0.68
St F, W, St

28. (0.3)& œ 0.002

CHAPTER 4 QUIZ
#' #& #% #$ ## #&$
29. a. &#
† &"
† &!
† %*
† %)
œ ***'
1. Falseß subjective probability can be
used when other types of probabilities b. "$
† "#
† ""
† "!
† *
œ $$
&# &" &! %* %) ''ß'%!
cannot be found.
2. False, empirical probability uses
c. 0
frequency distributions.
3. True !Þ$&
30. !Þ'& œ 0.538
4. False, P(A or B) œ P(A)  P(B)  P(A
and B) !Þ"'
5. False, the probabilities can be different. 31. !Þ$ œ 0.533
6. False, complementary events cannot !Þ&(
occur at the same time. 32. !Þ(
œ 0.814
7. True
!Þ!#)
8. False, order does not matter in 33. !Þ& œ 0.056
combinations.
" $
9. b 34. a. # b. (
10. b and d
11. d 35. 1  (0.45)' œ 0.992
12. b
13. c 36. 1  ( &' Ñ% œ 0.518
14. b
15. d

80
 Chapter 4 - Probability and Counting Rules

37. 1  (0.15)' œ 0.9999886

38. 2,646

39. 40,320

40. 1,365

41. 1,188,137,600; 710,424,000

42. 720

43. 33,554,432

44. 56
"
45. 4

3
46. "4

"#
47. &&

48.
PE B, BP, PE
BP GB B, BP, GB
B
MP PE B, MP, PE
GB B, MP, GB

PE P. BP, PE
BP GB P, BP, GB
P
MP PE P, MP, PE
GB P, MP, GB

PE C, BP, PE
BP GB C, BP, GB
C
MP PE C, MP, PE
GB C, MP, GB

PE V, BP, PE
BP GB V, BP, GB
V
MP PE V, MP, PE
GB V, MP, GB

81