Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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Exercises 8.4, page 459
1. Yes
3. No. There are more than two outcomes to each trial.
5. No. The probability of an accident on a clear day is not the same as
the probability of an accident on a rainy day.
7. .296 9. .051 11. .132
13. .656 15. .004 17. .116
19. a. P(X 0) ⬇ .078; P(X 1) ⬇ .259;
P(X 2) ⬇ .346; P(X 3) ⬇ .230;
P(X 4) ⬇ .077; P(X 5) ⬇ .010
b.
x 012345
P(X ⴝ x) .078 .259 .346 .230 .077 .010
c. 2; ⬇ 1.1
21. No. The probability that at most 1 is defective is
P(X 0) P(X 1) .74.
23. .165 25. .0002
27. a. .633 b. .367 29. a. .0329 b. .329
31. a. .273 b. .650 33. a. .817 b. .999
35. a. .075 b. .012 37. a. .122 b. .358
39. .349 41. At least 7 times 43. 375; ⬇ 9.68
45. False 47. False
Exercises 8.5, page 469
1. .9265 3. .0401 5. .8657
7. a. b. .9147
9. a. b. .2578
11. a. b. .8944
13. a. b. .2266
2.020.68
–1.25
–0.65
1.37
012345
4
3
2
1
.
.
.
.
15. a. 1.23 b. 0.81 17. a. 1.9 b. 1.9
19. a. .9772 b. .9192 c. .7333
Exercises 8.6, page 477
1. a. .2206 b. .2206 c. .2960
3. a. .0228 b. .0228 c. .4772 d. .7258
5. a. .0038 b. .0918 c. .4082 d. .2514
7. .6247 9. 0.62% 11. A: 80; B: 73; C: 62; D: 54
13. a. .4207 b. .4254 c. .0125
15. a. .2877 b. .0008 c. .7287
17. .9265 19. .8686
21. a. .0037 b. The drug is effective. 23. 2142
Chapter 8 Concept Review Questions, page 480
1. Random 2. Finite; infinite; continuous 3. Sum; .75
4. a. b.
5.
6. Fixed; two; same; independent
7. Continuous; probability density function; set
8. Normal; large; 0; 1
Chapter 8 Review Exercises, page 480
1. a. {WWW, BWW, WBW, WWB, BBW, BWB, WBB, BBB}
b.
Outcome WWW BWW WBW WWB
Value of X 0 111
Outcome BBW BWB WBB BBB
Value of X 2 223
c.
x 0123
P(X ⴝ x)
d.
2. $100
3. a. .8 b. 2.7; ⬇ 1.42
4. a.
x 01234
P(X ⴝ x) .1296 .3456 .3456 .1536 .0256
b. 1.6; Var(X ) 0.96; ⬇ 0.9798
x
0123
16
35
12
35
8
35
4
35
4
35
18
35
12
35
1
35
p
1
1x
1
m 2
2
p
2
1x
2
m 2
2
p
p
n
1x
n
m 2
2
; 2Var1X2
a
a b
P1E2
P1E
C
2
602 ANSWERS TO CHAPTER 8 ODD-NUMBERED EXERCISES
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 602
5. .6915
6. .2266
7. .4649
8. .4082
9. 2.42 10. 1.05 11. 2.03 12. 1.42 13. .6915
14. .8413 15. .2417 16. .7333 17. 41.3 mph
18. $12,000 19. .2646; .9163 20. 15.87% 21. At least .75
22. €21.54 billion; €0.5389 billion 23. 27.87; 6.41
24. .677 25. 120; ⬇ 10.1 26. 0.6%
27. a. .246 b. .901 28. a. .050 b. .995
29. .9738 30. .9997
Chapter 8 Before Moving On, page 481
1.
x 3 2 0123
P(X ⴝ x) .05 .1 .25 .3 .2 .1
2. a. .8 b. .92
3. 0.44; 4.0064; 2.0016
4. a. .2401; .4116; .2646; .0756; .0081 b. 1.2; .917
5. a. .9772 b. .9772 c. .9544
6. a. .0222 b. .6085 c. .0222
CHAPTER 9
Exercises 9.1, page 490
1. Yes 3. Yes 5. No 7. Ye s 9. No
11. a. Given that the outcome state 1 has occurred, the conditional prob-
ability that the outcome state 1 will occur is .3.
b. .7 c. c
.48
.52
d
−0.42 0.66
0.5– 0.75 0
−0.75
0.50
13. ;
15. 17.
19. a.
b. c. d. .85
21. a. Vote is evenly split. b. Democrat
23. After one year: 78.8% in the city and 21.2% in the suburbs. After two
years: 77.7% in the city and 22.3% in the suburbs.
25. University: 37%, Campus: 35%, Book Mart: 28%; University: 34.5%,
Campus: 31.35%, Book Mart: 34.15%
27. Business: 36.0%, Humanities: 23.8%, Education: 15.0%, Natural
Sciences and others: 25.1%
29. False
Using Technology Exercises 9.1, page 494
1.
3. Manufacturer A will have 23.95% of the market share, manufacturer
B will have 49.71% of the market share, and manufacturer C will
have 26.34% of the market share.
Exercises 9.2, page 500
1. Regular 3. Not regular 5. Regular
7. Not regular 9. 11.
13. 15. 17. 81.8%£
3
19
8
19
8
19
§£
3
13
8
13
2
13
§
c
2
7
5
7
dc
3
11
8
11
d
X
5
E
.204489
.131869
.261028
.186814
.215800
U
X
0
L
R
c
.5
.5
d
T
L
R
c
LR
.8 .9
.2 .1
d
State 1
State 1
State 2
.8
.2
.9
.1
State 2
State 1
State 2
Current
state
Next
state
.5
.5
X
2
£
5
16
27
64
17
64
§X
2
c
.576
.424
d
State 1
State 1
State 2
.6
.4
.2
.8
State 2
State 1
State 2
Current
state
Next
state
.5
.5
TX
0
c
.4
.6
d
ANSWERS TO CHAPTER 9 ODD-NUMBERED EXERCISES 603
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 603
19. 40.8% one wage earner and 59.2% two wage earners;
30% one wage earner and 70% two wage earners
21. 72.5% in single-family homes and 27.5% in condominiums;
70% in single-family homes and 30% in condominiums
23. a. 31.7% ABC, 37.35% CBS, 30.95% NBC
b. 33 % ABC, 33 % CBS, NBC
25. 25% red, 50% pink, 25% white 27. False
Using Technology Exercises 9.2, page 504
1.
Exercises 9.3, page 510
1. Yes 3. Yes 5. Yes 7. Ye s
9.
11. ,
13. ,
and , or
,
, , and so forth
15. 17.
19. 21.
23.
25. a.
UL L
UL
L
c
1.2
0.8
d, R 3.84, and S 3.24
E
100
7
22
3
11
010
5
22
9
22
001
5
11
7
22
00000
00000
U
≥
10
1
2
1
2
01
1
2
1
2
0000
0000
¥≥
1011
0100
0000
0000
¥
£
111
000
000
§c
11
00
d
S c
.4 .2
0.3
dR c
.4 .2
.2 .3
d
≥
10.4.2
01 0.3
00.4.2
00.2.3
¥
S c
.2 .4
.3 0
dR c
.3 .2
.2 .4
d
≥
10.2.4
01.3 0
00.3.2
00.2.4
¥
£
1.5.4
00.2
0.5.4
§, R c
0.2
.5 .4
d, and S 3.5
.44
R c
.4 .5
.2 0
d, and S 3.4
.54, or£
1.4.5
0.4.5
0.2 0
§
c
1.4
0.6
d, R 3.64, and S 3.44
X
5
E
.2045
.1319
.2610
.1868
.2158
U
33
1
3
%
1
3
1
3
b. ; eventually, only unleaded fuel will be used.
27. .25; .50; .75
29. a. DG 1 2
b.
c. .675
31. False
Exercises 9.4, page 519
1. R: row 1; C: column 2 3. R: row 1; C: column 1
5. R: row 1 or row 3; C: column 3
7. R: row 1 or row 3; C: column 2
9. Strictly determined;
a. 2 b. R: row 1; C: column 1
c. 2 d. Favors row player
11. Strictly determined;
a. 1 b. R: row 1; C: column 1
c. 1 d. Favors row player
13. Strictly determined;
a. 1 b. R: row 1; C: column 1
c. 1 d. Favors row player
15. Not strictly determined 17. Not strictly determined
19. a.
b. Robin: row 1; Cathy: column 1 or column 2
c. Not strictly determined
d. Not strictly determined
21. a. Economy
Good Recess.
Expand 200,000 120,000
Mgmt.
Not exp.
冤
50,000 150,000
冥
b. Yes
23. a. Charley
Raises Holds Lowers
Raises 3 1 3
Roland Holds 2 0 2
Lowers
冤
52 1
冥
25. True
Exercises 9.5, page 530
1. 3. 5. 0.16
5
12
3
10
£
2 34
345
4 56
§
≥
1 0 .325 .1
0 1 .675 .9
00 0 0
00 0 0
¥
D
G
1
2
≥
1 0 .25 .1
01 0.9
00 0 0
0 0 .75 0
¥
c
11
00
d
604 ANSWERS TO CHAPTER 9 ODD-NUMBERED EXERCISES
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 604
7. a. 1 b. 2 c. 0
d. .3; (a) is most advantageous
9. a. 1 b. .35 c. The first pair of strategies
11. P , Q , and E 2.5; favors row player
13. P , Q , and E ; favors column player
15. P , Q , and E 5; favors column player
17. a. P and Q
b. E 0; no
19. a. $5714 in hotel stocks; $34,286 in brewery stock
b. $4857
21. a. C
NF
N .48 .65 C Carlton; R Russell
R
F
冤
.50 .45
冥
N local newspaper; F flier
b. Russell’s strategy: P ⬇ [.23 .77]
Carlton’s strategy: Q ⬇
Chapter 9 Concept Review Questions, page 533
1. Probabilities; preceding 2. State; state
3. Transition 4. n n; nonnegative; 1
5. Distribution; steady-state
6. Regular; columns; equal; positive; TX X; elements; 1
7. Absorbing; leave; steps
8. a. Zero-sum b. Maximin; minimax 9. Optimal
10. Saddle point; maximin; row; saddle; minimax; column; saddle
Chapter 9 Review Exercises, page 534
1. Not regular 2. Regular
3. Regular 4. Not regular
5. 6.
7. Yes 8. No 9. No 10. Yes
11. 12.
13. 14. £
.323 .323 .323
.290 .290 .290
.387 .387 .387
§£
.457 .457 .457
.200 .200 .200
.343 .343 .343
§
c
4
9
4
9
5
9
5
9
dc
3
7
3
7
4
7
4
7
d
£
.1915
.4215
.387
§£
.3675
.36
.2725
§
c
.91
.09
d
c
1
3
2
3
d3
1
3
2
3
4
c
1
4
3
4
d3
1
2
1
2
4
1
7
c
5
7
2
7
d3
4
7
3
7
4
c
1
2
1
2
d3
1
4
3
4
4
15. a. AUN
b. c.
16. 12.5% large cars, 30.36% intermediate-sized cars, 57.14% small cars
17. Strictly determined; R: row 3; C: column 1; value is 4.
18. Strictly determined; R: row 1; C: column 2; value is 0.
19. Strictly determined; R: row 1; C: column 1; value is 1.
20. Not strictly determined
21. 22. 23. 2 24. 1.04
25. P , Q , and E ; favors row player
26. P , Q , and E , favors column player
27. P , Q , and E 1.2; favors row player
28. P , Q , and E 10.8; favors row player
29. a. b. $7
30. 25% compact models; 75% subcompact models
Chapter 9 Before Moving On, page 535
1. 2. 3.
4. a. 1 b. R: row 1; C; column 3 c. 1; column player
5. a. 3 units b. 1.22 units
6. a. P , Q b. ; row player
APPENDIX A
Exercises A.1, page 541
1. Yes 3. Yes 5. No
7. Yes 9. Yes 11. No
13. No 15. Negation 17. Conjunction
19. Conjunction
21. New orders for manufactured goods did not fall last month.
23. Drinking during pregnancy does not affect both the size and weight
of babies.
25. The commuter airline industry is not now undergoing a shakeup.
4
3
c
1
6
5
6
d3
2
3
1
3
4
£
111
000
000
§c
3
11
8
11
dc
.366
.634
d
c
.5 .7
.4 .5
d
c
2
5
3
5
d3
4
5
1
5
4
c
4
5
1
5
d3
1
10
9
10
4
1
2
c
13
22
9
22
d3
1
2
1
2
4
1
2
c
5
6
1
6
d3
1
2
1
2
4
10
9
1
4
A
U
N
£
.424
.262
.314
§
A
U
N
£
.50
.15
.35
§
A Agriculture
U Urban
N Nonagricultural
A
U
N
£
.85 0 .10
.10 .95 .05
.05 .05 .85
§
ANSWERS TO APPENDIX A ODD-NUMBERED EXERCISES 605
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 605
27. a. Domestic car sales increased over the past year, or foreign car
sales decreased over the past year, or both.
b. Domestic car sales increased over the past year, and foreign car
sales decreased over the past year.
c. Either domestic car sales increased over the past year or foreign
car sales decreased over the past year.
d. Domestic car sales did not increase over the past year.
e. Domestic car sales did not increase over the past year, or foreign
car sales decreased over the past year, or both.
f. Domestic car sales did not increase over the past year, or foreign
car sales did not decrease over the past year, or both.
29. a. Either the doctor recommended surgery to treat Sam’s hyperthy-
roidism or the doctor recommended radioactive iodine to treat
Sam’s hyperthyroidism.
b. The doctor recommended surgery to treat Sam’s hyperthyroidism,
or the doctor recommended radioactive iodine to treat Sam’s
hyperthyroidism, or both.
31. a. p q b. p Ɽ q c. ⬃p ⬃q d. ⬃(⬃q)
33. a. Both the popularity of prime-time soaps and prime-time situation
comedies did not increase this year.
b. The popularity of prime-time soaps did not increase this year, or
the popularity of prime-time detective shows decreased this year,
or both.
c. The popularity of prime-time detective shows decreased this year,
or the popularity of prime-time situation comedies did not increase
this year, or both.
d. Either the popularity of prime-time soaps did not increase this
year or the popularity of prime-time situation comedies did not
increase this year.
Exercises A.2, page 544
1.
pq⬃qp⬃q
TT F T
TF T T
FT F F
FF T T
3.
p ⬃p ⬃(⬃p)
TF T
FT F
5.
p ⬃pp⬃p
TF T
FT T
7.
pq⬃pp q ⬃p ( p q)
TTF T F
TFF T F
FTT T T
FFT F F
9.
pq⬃qp qp⬃q ( p q) (p ⬃q)
TT F T F F
TF T T T T
FT F T F F
FF T F F F
11.
pqp q ⬃( p q)(p q) ⬃( p q)
TT T F F
TF T F F
FT T F F
FF F T F
13.
pqrp qp r ( p q) (p r)
TTT T T T
TTF T T T
TFT T T T
TFF T T T
FTT T T T
FTF T F F
FFT F T F
FFF F F F
15.
pqrp q ⬃r ( p q) ⬃r
TTT T F T
TTF T T T
TFT F F F
TFF F T T
FTT F F F
FTF F T T
FFT F F F
FFF F T T
17.
pqr⬃qp⬃qp r ( p ⬃q) (p r)
TTT F F T T
TTF F F F F
TFT T T T T
TFF T T F T
FTT F F F F
FTF F F F F
FFT T F F F
FFF T F F F
19. 16 rows
Exercises A.3, page 548
1. ⬃q 씮 p; q 씮 ⬃p; ⬃p 씮 q
3. p 씮 q; ⬃p 씮 ⬃q; ⬃q 씮 ⬃p
5. Conditional: If it is snowing, then the temperature is below freezing.
Biconditional: It is snowing if and only if the temperature is below
freezing.
7. Conditional: If the company’s union and management reach a
settlement, then the workers do not strike.
Biconditional: The company’s union and management will reach a
settlement if and only if the workers do not strike.
9. False 11. False
13. It is false when I do not buy the house after the owner lowers the sell-
ing price.
15.
pqp씮 q ⬃( p 씮 q)
TT T F
TF F T
FT T F
FF T F
606 ANSWERS TO CHAPTER 1 ODD-NUMBERED EXERCISES
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 606
17.
pqp씮 q ⬃( p 씮 q) ⬃( p 씮 q) p
TT T F F
TF F T T
FT T F F
FF T F F
19.
pq⬃p ⬃qp씮 ⬃q ( p 씮 ⬃q) Ɽ⬃p
TT F F F F
TF F T T T
FT T F T F
FF T T T F
21.
pq⬃p ⬃qp씮 q ⬃q 씮 ⬃p ( p 씮 q) 씯씮 (⬃q 씮 ⬃p)
TT F F T T T
TF F T F F T
FT T F T T T
FF T T T T T
23.
pqp qp q ( p q) 씮 ( p q)
TT T T T
TF F T T
FT F T T
FF F F T
25.
pqrp q ⬃r ( p q) 씮 ⬃r
TTT T F F
TTF T T T
TFT T F F
TFF T T T
FTT T F F
FTF T T T
FFT F F T
FFF F T T
27.
pqrq rp씮 (q r)
TTT T T
TTF T T
TFT T T
TFF F F
FTT T T
FTF T T
FFT T T
FFF F T
29. Logically equivalent
31. Logically equivalent
33. Not logically equivalent
35. Not logically equivalent
37. a. p 씮 ⬃q b. ⬃p 씮 q c. ⬃q 씯씮 p
d. p 씮 ⬃q e. p 씯씮 ⬃q
Exercises A.4, page 552
1.
ppp p
TT T
FF F
3.
pqrp q
TTT T
TTF T
TFT F
TFF F
FTT F
FTF F
FFT F
FFF F
( p q) rq rp (q r)
TTT
FFF
FFF
FFF
FTF
FFF
FFF
FFF
5.
pqp qq p
TT T T
TF F F
FT F F
FF F F
7.
pqrq rp (q r)
TTT T T
TTF F T
TFT F T
TFF F T
FTT T T
FTF F F
FFT F F
FFF F F
p qp r ( p q) ( p r)
TT T
TT T
TT T
TT T
TT T
TF F
FT F
FF F
9. Tautology 11. Tautology 13. Tautology
15. Tautology 17. Neither
19. ⬃( p q): The candidate does not oppose changes in the Social
Security system, or the candidate does not support immi-
gration reform.
⬃( p q): The candidate does not oppose changes in the Social
Security system, and the candidate does not support
immigration reform.
ANSWERS TO APPENDIX A ODD-NUMBERED EXERCISES 607
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 607
21. [ p (q ⬃q) ( p q)]
⇔ p t ( p q)] By law 11
⇔ p ( p q) By law 14
23. ( p ⬃q) ( p ⬃r)
⇔ p (⬃q ⬃r) By law 7
25. ( p ⬃(q r)
⇔ p (⬃q ⬃r) By law 10
⇔ ( p ⬃q) ( p ⬃r) By law 7
Exercises A.5, page 557
1. Valid 3. Valid 5. Invalid 7. Valid
9. Valid 11. Valid 13. Valid 15. Invalid
17. p 씮 q; invalid 19. p q; valid
⬃p ⬃p 씮 ⬃q
⬃q p
21. p 씮 q; invalid 23. b
q 씮 r
r
p
25.
pqp씮 q ⬃q ⬃p
TT T F F
TF F T F
FT T F T
FF T T T
Exercises A.6, page 560
1. p q (r s)
3. [( p q) r] (⬃r p)
5. [( p q) r] (⬃p) [⬃q ( p r ⬃r)]
7.
9.
11.
13. p [⬃q (⬃p q)]; p ⬃q
15. p [⬃p q (q r)]; p q
APPENDIX C
Exercises, page 571
1. log
2
64 6 3. log
3
2 5. log
32
8
7. 1.0792 9. 1.2042 11. ln a
2
b
3
13. ln 15. x 8 17. x 10
19. x 21. x 23. 5.1986
25. 0.0912 27. 8.0472 29. 4.9041
11
3
3
2
31x y
1
3
z
3
5
1
9
B
p
A
~p
s
q
~q
r
r
B
q
p
A
~p
~r
p
A
q
B
r
608 ANSWERS TO APPENDIX C ODD-NUMBERED EXERCISES
87533_11_ANS_p579-608 1/31/08 8:53 AM Page 608
Abscissa, 3
Absorbing state, 505
Absorbing stochastic matrix, 505
Amortization, 287–289
formula, 287
schedule, 288
Annuity, 276
certain, 276
compound amount factor, 277
future value, 278
ordinary, 276
present value, 279
simple, 276
term, 276
Arithmetic progression, 300–302
common difference, 300
nth term, 300
sum, 301
Augmented matrix, 78
Average, 427
Axes, 2
Basic variables, 203
Bayes’ theorem, 402–405
Bernoulli trials, 452–459
Binding constraints, 194
Binomial distribution, 452–459
approximation by a normal distribution,
472–475
binomial trials, 453
mean, 457
standard deviation, 457
variance, 457
Binomial experiment, 453
Binomial random variable, 455
Binomial trials, 453
Birthday problem, 384–385
Book value, 30, 304
Bounded solution set, 160
Break-even analysis, 40–43
Break-even point, 41
Cartesian coordinate system, 2–4
abscissa, 3
axes, 2
ordered pair, 2
ordinate, 3
quadrants, 3
Central limit theorem, 475
Chebychev’s inequality, 444–446
Circle, 5
Column matrix, 102
Column vector, 102
Combinations, 340–342
Complement of a set, 317
Compound interest, 259–262. See also
Interest
Conditional probability, 388–393
Connectives, 539, 544–547
Continuous compound interest, 262–263
Continuous probability distribution, 462
Continuous random variable, 419, 462
Coordinates, 3
Corner point, 174
Cost function, 30
Counting problem, 323
Current state, 484
Demand curve, 31
Demand equation, 31
Demand function, 31
De Morgan’s laws, 318, 549
Dependent system, 70
Dependent variable, 28
Depreciation
book value, 30, 304
double-declining balance, 304–305
linear, 30
simple, 30
sum-of-the-years’-digits, 307
Deviations, 441
Disjoint sets, 317
Disjunction, 539
Distance formula, 3
Distribution vector, 488
Domain, 28
Dual problem, 228
Effective rate of interest, 263
Empty set, 315
Equation of a circle, 5
Equations of straight lines, 17
Equilibrium
price, 44
quantity, 44
Equivalent system, 76
Events, 354–356
independent, 395–398
mutually exclusive, 356, 363
probability of, 362–366
simple, 363
union of, 355
Expected value
of a binomial random variable, 457
of a game, 524
of a random variable, 428
Experiment, 354
event, 354
outcome, 354
sample point, 354
sample space, 354
Factorial, 337
Fair game, 432, 517, 522
Feasible set, 172
Feasible solution, 172, 203
Finite sample space, 354
Finite stochastic process, 393
Fixed costs, 31
Function, 27
cost, 31
demand, 31
dependent variable, 28
domain, 28
independent variable, 28
linear, 28
objective, 164
probability, 363
probability density, 462
profit, 30
range, 28
revenue, 30
Fundamental theorem of duality, 228
Future value, 265
Gambler’s ruin, 506–508
Game theory, 512–518, 521–529
fair game, 432, 522
games with mixed strategies, 521–528
expected value, 524–526
optimal strategies, 516
maximin strategy, 514
minimax strategy, 514
mixed strategy, 522
nonstrictly determined game, 526
optimal strategy, 516
payoff matrix, 513
pure strategy, 516
saddle point, 516
strictly determined game, 516
two-person games, 512–513
value of a game, 523
zero-sum game, 512
Gauss–Jordan elimination method, 75–85,
91–96
equivalent system, 76
row operations, 80
steps in, 82
Generalized multiplication principle, 332
Genetics, 508–509
Geometric progression, 302–304
common ratio, 302
nth term, 302
sum, 304
Graphing linear inequalities, 156–161
Half-planes, 156
Histogram, 421–422
INDEX
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610 INDEX
Identity matrix, 117
Inconsistent system, 70
Independent events, 395–398
Independent variable, 28
Input–output analysis, 141–146
Intercepts, 15
Interest
compound, 259–262
formula, 260
future value, 265
present value, 265
principal, 260
continuous compound, 262–263
conversion period, 259
rate
effective, 263
nominal, 259
simple, 258–259
accumulated amount, 258
formula, 258
Internal consumption matrix, 143
Intersection of lines, 40–46
Intersection of sets, 317
Inverse of a matrix, 128
Irrational number, 564
Isoprofit line, 173
Least-squares line, 52
Leontief, Wassily, 67, 141
Leontief input–output model, 141–146
input–output matrix, 142
internal consumption matrix, 143
Linear depreciation, 30–31
Linear equations, 12–17
general form, 17
in n variables, 72
intercepts, 15
point-slope form, 13
slope, 10
slope-intercept form, 15
vertical lines, 12, 17
Linear function(s), 28–33
break-even analysis, 40–43
demand and supply curves, 31–33
simple depreciation, 30–31
Linear inequality, 156–161
Linear programming
corner point, 174
feasible set, 172
feasible solution, 172, 203
graphical solution, 172–178
linear constraints, 172
method of corners, 174
objective function, 164
optimal solution, 172
problem, 164, 202
basic solution, 204
dual, 228
existence of a solution, 174
nonstandard, 242–250
primal, 228
standard, 202, 226, 242
simplex method, 202–216, 226–234,
242–250
basic variables, 202
nonbasic variables, 203
nonstandard problems, 242
optimal solution(s), 202
pivot column, 205
pivot element, 205
pivot row, 205
simplex tableau, 206
slack variables, 202
standard maximization problem, 202
standard minimization problem, 227
steps in, 207, 246
theorem, 174
Lines. See Linear equations
Logarithms, 567–571
Logic, 537–561
argument, 533–554
conclusion, 554, 553
conditional statement, 544
converse, 545
contrapositive, 545
inverse, 545
connectives, 539
biconditional, 547
conditional, 544
conjunction, 539
disjunction, 539
exclusive disjunction, 540
inclusive disjunction, 539
negation, 540
contradiction, 550
hypothesis, 544
laws, 549–551
logical equivalence, 546, 551
logical variants, 545
order of precedence, 548
proposition, 553
switching networks, 558
tautology, 550
truth tables, 542–544
Market equilibrium, 43–46
Markov chains, 484–489
absorbing, 505
state, 505
stochastic matrix, 505
current state, 484
regular, 496
steady-state distribution vector, 495
steady-state matrix, 496
stochastic matrix
absorbing, 505
regular, 486
transition matrix, 485
transition probabilities, 485
Mathematical model, 26
Matrix, 102
absorbing stochastic, 505
addition, 105
associative law, 105
commutative law, 105
augmented, 78
coefficient, 78
column, 102
cost, 118
dimension, 102
elements, 102
entries, 102
equality, 103
equation, 106
identity, 117
ijth entry (a
ij
), 102
input–output, 142
internal consumption, 143
inverse, 128
inverse of a 2 ⫻ 2 matrix, 131
multiplication, 113–119
associative law, 117
distributive law, 117
nonsingular, 128
pivot element, 81
product, 114
regular stochastic, 486
row, 102
row-reduced form, 79
scalar product, 106
singular, 128
size, 102
square, 102
steady-state, 496
subtraction, 104
transition, 486
transpose, 106
zero, 105
Maximin strategy, 514
Mean, 427
Median, 435
Method of corners, 174
Method of least squares, 51–55
least-squares line, 52
normal equations, 53
principle, 52
regression line, 52
scatter diagram, 52
Minimax strategy, 514
Mixed constraints, 242
Mixed strategy, 522
Mode, 435
Modeling, 26–27
Multiplication principle, 329–333
Mutually exclusive events, 356, 363
Natural numbers, 564
Nonbasic variables, 203
Nonstandard linear programming problems,
242–250
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Normal approximation of binomial
distribution, 472–475
Normal curve, 463
Normal distribution, 463
Normal equations, 52
Number line, 2
Objective function, 164
Odds, 432–433
Optimal solution, 172
Optimal strategy, 514–519, 526
Ordered pair, 2
Ordinary annuity, 276
Ordinate, 3
Origin, 2
Outcome, 354
Parallel lines, 12
Parameter, 69
Partition, 403
Permutations, 335–340
Perpendicular lines, 14
Pivot column, 205
Pivot element, 205
Pivoting, 81
Pivot row, 205
Point-slope form, 13
Present value, 265
Primal problem, 228
Principle of least squares, 52
Probability
addition principle, 365
a posteriori, 402
a priori, 402
Bayes’ theorem, 403
Bernoulli trials, 452–459
binomial distribution, 455
binomial experiment, 453
conditional, 388–393
continuous probability distribution, 462
counting techniques, 329–333, 381–384
density function, 463
distribution, 363, 419–420
elementary event, 363
empirical, 363
of an event, 362–365
expected value, 428
experiment, 354
event, 354
outcome, 354
finite stochastic process, 393
function, 363
independent event, 395
mutually exclusive events, 356, 363
partition, 403
product rule, 391
properties, 371–374
relative frequency, 363
rule of complements, 374
sample point, 354
INDEX 611
sample space, 354
event, 354
finite, 354
partition, 403
reduced, 389
uniform, 364
simple event, 363
transition, 485
tree diagram, 303, 339, 356–358,
393–395
Product rule, 391
Profit function, 30
Pure strategy, 522
Quadrant, 3
Random variable, 418
binomial, 455
continuous, 419, 462
expected value, 428
finite discrete, 419
infinite discrete, 419
probability distribution of, 419
standard deviation of, 443
standard normal, 464
variance, 441
Range, 28
Rational numbers, 564
Real number line, 2
Real number system, 564–565
integers, 564
irrational numbers, 564
natural numbers, 564
rational numbers, 564
whole numbers, 564
Regular Markov chain, 496
Regular stochastic matrix, 498
Relative frequency, 363
Revenue function, 30
Roster notation, 314
Roulette, 431
Row matrix, 102
Row operations, 80–81
Row-reduced form, 79
Saddle point, 516
Sample space, 354
finite, 354
reduced, 389
uniform, 364
Scalar, 106
Scalar product, 106
Scatter diagram, 52
Sensitivity analysis, 185–194
binding constraints, 194
changes in the objective function
coefficients, 186–189
changes in the right-hand side constants,
189–191
shadow prices, 191
Set(s), 314–319
complement, 317
definition, 314
De Morgan’s laws, 318, 549
disjoint, 317
elements, 314
empty, 315
equal, 314
intersection, 317
members, 314
notation
roster, 314
set-builder, 314
number of elements in, 323–325
operations, 316–319
proper subset, 315
subset, 315
union, 317
universal, 316
Venn diagrams, 316–319
Shadow price, 191–194
Simple depreciation, 30
Simple interest, 258
Simplex method. See Linear programming
Simplex tableau, 206
Singular matrix, 128
Sinking fund, 291
Slack variables, 202
Slope, 10
Slope-intercept form, 15
Solution set of a system of linear
inequalities, 159
bounded, 160
unbounded, 160
Square matrix, 102
Standard deviation, 443
Standard linear programming problem, 202
Standard normal curve, 464
Standard normal variable, 464
Standard viewing window, 23
Steady-state distribution vector, 495
Steady-state matrix, 496
Stochastic matrix, 486
Stochastic process, 393, 484
Straight line depreciation, 30–31
Strategies. See Game theory
Strictly determined game, 516
Subset, 315
Supply curve, 32
Supply equation, 32
Supply function, 32
Switching networks, 558
Symbolic logic, 537
Systems of linear equations, 68–72
dependent, 70
equivalent, 76
Gauss–Jordan elimination method,
75–85, 91–96
inconsistent, 70
solution by inverses, 132–134
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