Solution Manual For
Introduction to Probability Models,12th Edition by Sheldon M. Ross
Chapter 1-11
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communicatio
Chapter 1
1. S = { (R, R), (R, G),(R, B), (G, R), (G, G),(G, B), (B, R), (B, G),(B, B) }
The probability of each point in S is 1/9.
2. S = {(R, G),(R, B), (G, R), (G, B), (B, R), (B, G)}
3. S = {(e1, e 2 ,. .., en), n ≥ 2} where ei ∈(heads, tails}. In addition, en = en—1 =
heads and for i = 1 , . . . , n — 2 if ei = heads, then ei+1 = tails.
P{4 tosses}= P{(t, t, h, h)}+ P{(h, t, h, h)}
1 4 1
=2 =
2 8
4. (a) F(E ∪ G)c = FE c G c
(b) EFG c
(c) E F G
(d) EF EG FG
(e) EFG
(f) (E ∪ F ∪ G)c = E c F c G c
(g) (EF) c (EG) c (FG) c
(h) (EFG) c
5. 34 . If he wins, he only wins $1, while if he loses, he loses $3.
6. If E(F G) occurs, then E occurs and either F or G occur; therefore, either EF
or EG occurs and so
E(F ∪ G) ⊂ EF ∪ EG
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communication
,2
Similarly, if EF ∪ EG occurs, then either EF or EG occurs. Thus, E occurs and
either F or G occurs; and so E(F ∪ G) occurs. Hence,
EF ∪ EG ⊂ E(F ∪ G)
which together with the reverse inequality proves the result.
7. If (E F)c occurs, then E F does not occur, and so E does not occur (and so Ec
does); F does not occur (and so Fc does) and thus Ec and Fc both occur. Hence,
(E ∪ F)c ⊂ Ec Fc
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business
terminology, theories, and principles.Case Studies: A staple of business exams, case studies present students with real-world
business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to
think critically and make strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to
explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business ExamsCritical Thinking
and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-
life situations. These tests assess decision-making skills, as well as the ability to evaluate various business
alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform
calculations and interpret data. The ability to analyze financial statements, project revenues, and manage budgets is
essential.Communication
If Ec Fc occurs, then Ec occurs (and so E does not), and Fc occurs (and so F does
not). Hence, neither E or F occurs and thus (E ∪ F)c does. Thus,
Ec Fc ⊂ (E ∪ F)c
and the result follows.
8. 1 ≥ P(E ∪ F) = P(E) + P(F) — P(EF)
9. F = E ∪ FE c, implying since E and FE are disjoint that P(F) = P(E) +
c
P(FE) c .
10. Either by induction or use
n
c c c c c
∪ Ei = E1 ∪ E1 E2 ∪ E1 E2 E3 ∪ · · · ∪ E1 · · · En—1 En
1
and as each of the terms on the right side are mutually exclusive:
P( ∪ Ei ) = P(E 1 ) + P(E c E2) + P(E c Ec E3) + · · ·
i 1 1 2
c c
+ P(E 1 · · · En—1 En)
≤ P(E 1 ) + P(E 2 ) + · · · + P(E n ) (why?)
36
11. P{sum is i } =
i = 8 , . . . , 12
13—i
36 ,
12. Either use hint or condition on initial outcome as:
P{E before F }
= P{E before F |initial outcome is E }P(E)
+ P{E before F |initial outcome is F }P(F)
+ P{E before F |initial outcome neither E or F }[1 — P(E) — P(F)]
= 1 · P(E) + 0 · P(F) + P{E before F }
= [1 — P(E) — P(F)]
13. Condition an initial toss
Therefore, P{E before F } = P
, 3
P(E) (E)+ P(F)
12
P{win}= P{win|throw i }P{throw i }
i =2
include:ultiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communication
Now,
P{win|throw i } = P{i before 7}
, 10 i = 2, 12
, i
⎨ —
,
i = 3,..., 6
=, 5 1
13 i
+
19 — 11 i = 7, 11
where above is obtained by using Problems 11 and 12.
P{win}≈ .49.
∞
14. P{ A wins}= P{ A wins on (2n + 1)st toss}
n=0
∞
= (1 — P) P
n=0
∞
=P [(1 — P) ]
n=0
1
=P
1 — (1 — P)2
P
=
2 P P2
1
=
2—P
P{B wins}= 1 — P{ A wins}
1— P
=
2— P
16. P(E ∪ F) = P(E ∪ FE c )
= P(E) + P(FE )
c
since E and FE c are disjoint. Also,
P(E) = P(FE ∪ FE c )
c
, 4
= P(FE) + P(FE ) by disjointness
Hence,
P(E ∪ F) = P(E) + P(F) — P(EF)
17. Prob{end}= 1 — Prob{continue}
= 1 — P({H, H, H } ∪ {T, T, T })
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology,
theories, and principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios
and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These
1 1 1 1 1 1
t Fair coin: Prob{end}= 1 —
· · + · ·
2 2 2 2 2 2
3
=
4 1 1 1 3 3 3
Biased coin: P{end}= 1 —
· · + · ·
4 4 4 4 4 4
9
=
16
18. Let B = event both are girls; E = event oldest is girl; L = event at least one is a girl.
P(BE) P(B) 1/4 1
(a) P(B|E) = = = =
P(E) P(E) 2
1 3
(b) P(L) = 1 — P(no girls) = 1 ,
— =
4 4
P(BL) P(B) 1/4 1
P(B|L) = = = =
P(L)
ypes of questions test the student’s ability analyze business concepts in3a detailed and coherent manner.1.3. Skills Tested in Business
to explain and P(L)
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-
life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For
subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze financial
statements, project revenues, and manage budgets is essential.Communication
19. E = event at least 1 six P(E)
number of ways to get E 11
= =
number of samples pts 36
D = event two faces are different P(D)
= 1 — Prob(two faces the same)
6 5 P(E D) 10/36 1
=1— = P(E |D) = = =
36 6 P(D) 5/6 3
20. Let E = event same number on exactly two of the dice; S = event all three numbers
are the same; D = event all three numbers are different. These three events are
mutually exclusive and define the whole sample space. Thus, 1= P(D)+ P(S) +
P(E), P(S) = = for D have six possible values for first die, five for
6/216 1/36;
second, and four for third.
∴ Number of ways to get D = 6 · 5 · 4 = 120.
Introduction to Probability Models,12th Edition by Sheldon M. Ross
Chapter 1-11
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communicatio
Chapter 1
1. S = { (R, R), (R, G),(R, B), (G, R), (G, G),(G, B), (B, R), (B, G),(B, B) }
The probability of each point in S is 1/9.
2. S = {(R, G),(R, B), (G, R), (G, B), (B, R), (B, G)}
3. S = {(e1, e 2 ,. .., en), n ≥ 2} where ei ∈(heads, tails}. In addition, en = en—1 =
heads and for i = 1 , . . . , n — 2 if ei = heads, then ei+1 = tails.
P{4 tosses}= P{(t, t, h, h)}+ P{(h, t, h, h)}
1 4 1
=2 =
2 8
4. (a) F(E ∪ G)c = FE c G c
(b) EFG c
(c) E F G
(d) EF EG FG
(e) EFG
(f) (E ∪ F ∪ G)c = E c F c G c
(g) (EF) c (EG) c (FG) c
(h) (EFG) c
5. 34 . If he wins, he only wins $1, while if he loses, he loses $3.
6. If E(F G) occurs, then E occurs and either F or G occur; therefore, either EF
or EG occurs and so
E(F ∪ G) ⊂ EF ∪ EG
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communication
,2
Similarly, if EF ∪ EG occurs, then either EF or EG occurs. Thus, E occurs and
either F or G occurs; and so E(F ∪ G) occurs. Hence,
EF ∪ EG ⊂ E(F ∪ G)
which together with the reverse inequality proves the result.
7. If (E F)c occurs, then E F does not occur, and so E does not occur (and so Ec
does); F does not occur (and so Fc does) and thus Ec and Fc both occur. Hence,
(E ∪ F)c ⊂ Ec Fc
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business
terminology, theories, and principles.Case Studies: A staple of business exams, case studies present students with real-world
business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to
think critically and make strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to
explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business ExamsCritical Thinking
and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-
life situations. These tests assess decision-making skills, as well as the ability to evaluate various business
alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform
calculations and interpret data. The ability to analyze financial statements, project revenues, and manage budgets is
essential.Communication
If Ec Fc occurs, then Ec occurs (and so E does not), and Fc occurs (and so F does
not). Hence, neither E or F occurs and thus (E ∪ F)c does. Thus,
Ec Fc ⊂ (E ∪ F)c
and the result follows.
8. 1 ≥ P(E ∪ F) = P(E) + P(F) — P(EF)
9. F = E ∪ FE c, implying since E and FE are disjoint that P(F) = P(E) +
c
P(FE) c .
10. Either by induction or use
n
c c c c c
∪ Ei = E1 ∪ E1 E2 ∪ E1 E2 E3 ∪ · · · ∪ E1 · · · En—1 En
1
and as each of the terms on the right side are mutually exclusive:
P( ∪ Ei ) = P(E 1 ) + P(E c E2) + P(E c Ec E3) + · · ·
i 1 1 2
c c
+ P(E 1 · · · En—1 En)
≤ P(E 1 ) + P(E 2 ) + · · · + P(E n ) (why?)
36
11. P{sum is i } =
i = 8 , . . . , 12
13—i
36 ,
12. Either use hint or condition on initial outcome as:
P{E before F }
= P{E before F |initial outcome is E }P(E)
+ P{E before F |initial outcome is F }P(F)
+ P{E before F |initial outcome neither E or F }[1 — P(E) — P(F)]
= 1 · P(E) + 0 · P(F) + P{E before F }
= [1 — P(E) — P(F)]
13. Condition an initial toss
Therefore, P{E before F } = P
, 3
P(E) (E)+ P(F)
12
P{win}= P{win|throw i }P{throw i }
i =2
include:ultiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and
principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios and ask them to apply their
knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic decisions.Essay/Short Answer
Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills
Tested in Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical
knowledge to real-life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative
Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze
financial statements, project revenues, and manage budgets is essential.Communication
Now,
P{win|throw i } = P{i before 7}
, 10 i = 2, 12
, i
⎨ —
,
i = 3,..., 6
=, 5 1
13 i
+
19 — 11 i = 7, 11
where above is obtained by using Problems 11 and 12.
P{win}≈ .49.
∞
14. P{ A wins}= P{ A wins on (2n + 1)st toss}
n=0
∞
= (1 — P) P
n=0
∞
=P [(1 — P) ]
n=0
1
=P
1 — (1 — P)2
P
=
2 P P2
1
=
2—P
P{B wins}= 1 — P{ A wins}
1— P
=
2— P
16. P(E ∪ F) = P(E ∪ FE c )
= P(E) + P(FE )
c
since E and FE c are disjoint. Also,
P(E) = P(FE ∪ FE c )
c
, 4
= P(FE) + P(FE ) by disjointness
Hence,
P(E ∪ F) = P(E) + P(F) — P(EF)
17. Prob{end}= 1 — Prob{continue}
= 1 — P({H, H, H } ∪ {T, T, T })
include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology,
theories, and principles.Case Studies: A staple of business exams, case studies present students with real-world business scenarios
and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make
strategic decisions.Essay/Short Answer Questions: These
1 1 1 1 1 1
t Fair coin: Prob{end}= 1 —
· · + · ·
2 2 2 2 2 2
3
=
4 1 1 1 3 3 3
Biased coin: P{end}= 1 —
· · + · ·
4 4 4 4 4 4
9
=
16
18. Let B = event both are girls; E = event oldest is girl; L = event at least one is a girl.
P(BE) P(B) 1/4 1
(a) P(B|E) = = = =
P(E) P(E) 2
1 3
(b) P(L) = 1 — P(no girls) = 1 ,
— =
4 4
P(BL) P(B) 1/4 1
P(B|L) = = = =
P(L)
ypes of questions test the student’s ability analyze business concepts in3a detailed and coherent manner.1.3. Skills Tested in Business
to explain and P(L)
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-
life situations. These tests assess decision-making skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For
subjects like finance or economics, business exams often require students to perform calculations and interpret data. The ability to analyze financial
statements, project revenues, and manage budgets is essential.Communication
19. E = event at least 1 six P(E)
number of ways to get E 11
= =
number of samples pts 36
D = event two faces are different P(D)
= 1 — Prob(two faces the same)
6 5 P(E D) 10/36 1
=1— = P(E |D) = = =
36 6 P(D) 5/6 3
20. Let E = event same number on exactly two of the dice; S = event all three numbers
are the same; D = event all three numbers are different. These three events are
mutually exclusive and define the whole sample space. Thus, 1= P(D)+ P(S) +
P(E), P(S) = = for D have six possible values for first die, five for
6/216 1/36;
second, and four for third.
∴ Number of ways to get D = 6 · 5 · 4 = 120.
