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Title:
Mathematics of chance
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Publication Information:
New York : John Wiley & Sons, 2001
ISBN:
9780471410898

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30000004879700 QA273.25 A54 2001 Open Access Book Book
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30000004879742 QA273.25 A54 2001 Open Access Book Book
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Summary

Summary

Mathematics of Chance utilizes simple, real-world problems-some of which have only recently been solved-to explain fundamental probability theorems, methods, and statistical reasoning. Jiri Andel begins with a basic introduction to probability theory and its important points before moving on to more specific sections on vital aspects of probability, using both classic and modern problems.
Each chapter begins with easy, realistic examples before covering the general formulations and mathematical treatments used. The reader will find ample use for a chapter devoted to matrix games and problem sets concerning waiting, probability calculations, expectation calculations, and statistical methods. A special chapter utilizes problems that relate to areas of mathematics outside of statistics and considers certain mathematical concepts from a probabilistic point of view. Sections and problems cover topics including:
* Random walks
* Principle of reflection
* Probabilistic aspects of records
* Geometric distribution
* Optimization
* The LAD method, and more
Knowledge of the basic elements of calculus will be sufficient in understanding most of the material presented here, and little knowledge of pure statistics is required. Jiri Andel has produced a compact reference for applied statisticians working in industry and the social and technical sciences, and a book that suits the needs of students seeking a fundamental understanding of probability theory.


Author Notes

Jiri Andel is Vice Dean of the Faculty of Mathematics and Physics at Charles University, Prague, Czech Republic


Reviews 1

Choice Review

Originally a Czech preprint, this compact reference book (the first book by Andel, of Charles Univ., Czech Republic, to be translated into English) provides a fundamental understanding of probability through the use of real-world problems. It can also be used as a guide for applied statisticians in industry and the sciences. Most of the book deals with probability concepts that are applied to numerous situations such as random walk, roulette, records, problems concerning waiting and optimization, expectation, and the Bernoulli problem. An entire chapter is devoted to matrix games and probability in mathematics covering such areas as quadratic equations, the Chebyshev problem, and lattice-point triangles. The book also contains more than 50 tables providing information for these real-world examples. Basic calculus and some understanding of statistics are required. Recommended for undergraduate-level library collections. Lower-division undergraduates; professionals. D. J. Gougeon University of Scranton


Table of Contents

Prefacep. xvii
Acknowledgmentsp. xix
Introductionp. xxi
1 Probabilityp. 1
1.1 Introductionp. 1
1.2 Classical probabilityp. 3
1.3 Geometric probabilityp. 8
1.4 Dependence and independencep. 10
1.5 Bayes' theoremp. 13
1.6 Medical diagnosticsp. 15
1.7 Random variablesp. 19
1.8 Mang Kung dice gamep. 22
1.9 Some discrete distributionsp. 24
1.10 Some continuous distributionsp. 26
2 Random walkp. 29
2.1 Gambler's ruinp. 29
2.2 American roulettep. 32
2.3 A reluctant random walkp. 34
2.4 Random walk until no shoes are availablep. 38
2.5 Three-tower problemp. 39
2.6 Gambler's ruin problem with tiesp. 41
2.7 Problem of prize divisionp. 43
2.8 Tennisp. 47
2.9 Wolf and sheepp. 50
3 Principle of reflectionp. 53
3.1 Ticket-selling automatp. 53
3.2 Known structure of the queuep. 54
3.3 Queue with random structurep. 57
3.4 Random number of customersp. 59
4 Recordsp. 63
4.1 Records, probability, and statisticsp. 63
4.2 Expected number of recordsp. 64
4.3 Probability of r recordsp. 66
4.4 Stirling numbersp. 68
4.5 Indicatorsp. 70
4.6 When records occurp. 72
4.7 Temperature records in Praguep. 74
4.8 How long we wait for the next recordp. 75
4.9 Some applications of recordsp. 78
5 Problems that concern waitingp. 79
5.1 Geometric distributionp. 79
5.2 Problem about keysp. 83
5.3 Collection problemsp. 84
5.4 When two players wait for a successp. 86
5.5 Waiting for a series of identical eventsp. 86
5.6 Lunchp. 87
5.7 Waiting studentp. 89
5.8 Waiting for a bus in a townp. 90
6 Problems that concern optimizationp. 93
6.1 Analysis of bloodp. 93
6.2 Overbooking airline flightsp. 95
6.3 Secretary problemp. 97
6.4 A birthday is not a workdayp. 100
6.5 Votingp. 100
6.6 Dice without transitivityp. 103
6.7 How to increase reliabilityp. 105
6.8 Exam taking strategyp. 110
6.9 Two unknown numbersp. 113
6.10 Archersp. 115
6.11 A stacking problemp. 115
6.12 No risk, no winp. 117
7 Problems on calculating probabilityp. 119
7.1 Dormitoryp. 119
7.2 Too many marriagesp. 123
7.3 Tossing coins until all show headsp. 124
7.4 Anglersp. 126
7.5 Birdsp. 127
7.6 Sultan and Caliphp. 129
7.7 Penaltiesp. 130
7.8 Two 6s and two 5sp. 131
7.9 Principle of inclusion and exclusionp. 132
7.10 More heads on coinsp. 133
7.11 How combinatorial identities are bornp. 134
7.12 Examsp. 135
7.13 Wyvernsp. 136
7.14 Gaps among ballsp. 137
7.15 Numbered pegsp. 139
7.16 Crux Mathematicorump. 140
7.17 Ties in electionsp. 141
7.18 Crapsp. 143
7.19 Problem of exceeding 12p. 144
8 Problems on calculating expectationp. 147
8.1 Christmas partyp. 147
8.2 Spaghettip. 149
8.3 Elevatorp. 151
8.4 Matching pairs of socksp. 152
8.5 A guessing gamep. 153
8.6 Expected number of drawsp. 155
8.7 Length of the wirep. 156
8.8 Ancient Jewish gamep. 158
8.9 Expected value of the smallest elementp. 160
8.10 Ballot countp. 161
8.11 Bernoulli problemp. 164
8.12 Equal numbers of heads and tailsp. 165
8.13 Pearlsp. 166
9 Problems on statistical methodsp. 169
9.1 Proofreadingp. 169
9.2 How to enhance the accuracy of a measurementp. 171
9.3 How to determine the area of a squarep. 172
9.4 Two routes to the airportp. 175
9.5 Christmas inequalityp. 178
9.6 Cinderellap. 180
10 The LAD methodp. 185
10.1 Medianp. 185
10.2 Least squares methodp. 186
10.3 LAD methodp. 188
10.4 Laplace methodp. 189
10.5 General straight linep. 190
10.6 LAD method in a general casep. 191
11 Probability in mathematicsp. 195
11.1 Quadratic equationsp. 195
11.2 Sum and product of random numbersp. 198
11.3 Socks and number theoryp. 201
11.4 Tshebyshev problemp. 203
11.5 Random trianglep. 205
11.6 Lattice-point trianglesp. 209
12 Matrix gamesp. 211
12.1 Linear programmingp. 211
12.2 Pure strategiesp. 213
12.3 Mixed strategiesp. 215
12.4 Solution of matrix gamesp. 216
12.5 Solution of 2 [times] 2 gamesp. 218
12.6 Two-finger morap. 219
12.7 Three-finger morap. 220
12.8 Problem of colonel Blottop. 220
12.9 Scissors--paper--stonep. 221
12.10 Birthdayp. 222
Referencesp. 223
Topic Indexp. 231
Author Indexp. 234
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