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Searching... | 30000004879700 | QA273.25 A54 2001 | Open Access Book | Book | Searching... |
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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
Preface | p. xvii |
Acknowledgments | p. xix |
Introduction | p. xxi |
1 Probability | p. 1 |
1.1 Introduction | p. 1 |
1.2 Classical probability | p. 3 |
1.3 Geometric probability | p. 8 |
1.4 Dependence and independence | p. 10 |
1.5 Bayes' theorem | p. 13 |
1.6 Medical diagnostics | p. 15 |
1.7 Random variables | p. 19 |
1.8 Mang Kung dice game | p. 22 |
1.9 Some discrete distributions | p. 24 |
1.10 Some continuous distributions | p. 26 |
2 Random walk | p. 29 |
2.1 Gambler's ruin | p. 29 |
2.2 American roulette | p. 32 |
2.3 A reluctant random walk | p. 34 |
2.4 Random walk until no shoes are available | p. 38 |
2.5 Three-tower problem | p. 39 |
2.6 Gambler's ruin problem with ties | p. 41 |
2.7 Problem of prize division | p. 43 |
2.8 Tennis | p. 47 |
2.9 Wolf and sheep | p. 50 |
3 Principle of reflection | p. 53 |
3.1 Ticket-selling automat | p. 53 |
3.2 Known structure of the queue | p. 54 |
3.3 Queue with random structure | p. 57 |
3.4 Random number of customers | p. 59 |
4 Records | p. 63 |
4.1 Records, probability, and statistics | p. 63 |
4.2 Expected number of records | p. 64 |
4.3 Probability of r records | p. 66 |
4.4 Stirling numbers | p. 68 |
4.5 Indicators | p. 70 |
4.6 When records occur | p. 72 |
4.7 Temperature records in Prague | p. 74 |
4.8 How long we wait for the next record | p. 75 |
4.9 Some applications of records | p. 78 |
5 Problems that concern waiting | p. 79 |
5.1 Geometric distribution | p. 79 |
5.2 Problem about keys | p. 83 |
5.3 Collection problems | p. 84 |
5.4 When two players wait for a success | p. 86 |
5.5 Waiting for a series of identical events | p. 86 |
5.6 Lunch | p. 87 |
5.7 Waiting student | p. 89 |
5.8 Waiting for a bus in a town | p. 90 |
6 Problems that concern optimization | p. 93 |
6.1 Analysis of blood | p. 93 |
6.2 Overbooking airline flights | p. 95 |
6.3 Secretary problem | p. 97 |
6.4 A birthday is not a workday | p. 100 |
6.5 Voting | p. 100 |
6.6 Dice without transitivity | p. 103 |
6.7 How to increase reliability | p. 105 |
6.8 Exam taking strategy | p. 110 |
6.9 Two unknown numbers | p. 113 |
6.10 Archers | p. 115 |
6.11 A stacking problem | p. 115 |
6.12 No risk, no win | p. 117 |
7 Problems on calculating probability | p. 119 |
7.1 Dormitory | p. 119 |
7.2 Too many marriages | p. 123 |
7.3 Tossing coins until all show heads | p. 124 |
7.4 Anglers | p. 126 |
7.5 Birds | p. 127 |
7.6 Sultan and Caliph | p. 129 |
7.7 Penalties | p. 130 |
7.8 Two 6s and two 5s | p. 131 |
7.9 Principle of inclusion and exclusion | p. 132 |
7.10 More heads on coins | p. 133 |
7.11 How combinatorial identities are born | p. 134 |
7.12 Exams | p. 135 |
7.13 Wyverns | p. 136 |
7.14 Gaps among balls | p. 137 |
7.15 Numbered pegs | p. 139 |
7.16 Crux Mathematicorum | p. 140 |
7.17 Ties in elections | p. 141 |
7.18 Craps | p. 143 |
7.19 Problem of exceeding 12 | p. 144 |
8 Problems on calculating expectation | p. 147 |
8.1 Christmas party | p. 147 |
8.2 Spaghetti | p. 149 |
8.3 Elevator | p. 151 |
8.4 Matching pairs of socks | p. 152 |
8.5 A guessing game | p. 153 |
8.6 Expected number of draws | p. 155 |
8.7 Length of the wire | p. 156 |
8.8 Ancient Jewish game | p. 158 |
8.9 Expected value of the smallest element | p. 160 |
8.10 Ballot count | p. 161 |
8.11 Bernoulli problem | p. 164 |
8.12 Equal numbers of heads and tails | p. 165 |
8.13 Pearls | p. 166 |
9 Problems on statistical methods | p. 169 |
9.1 Proofreading | p. 169 |
9.2 How to enhance the accuracy of a measurement | p. 171 |
9.3 How to determine the area of a square | p. 172 |
9.4 Two routes to the airport | p. 175 |
9.5 Christmas inequality | p. 178 |
9.6 Cinderella | p. 180 |
10 The LAD method | p. 185 |
10.1 Median | p. 185 |
10.2 Least squares method | p. 186 |
10.3 LAD method | p. 188 |
10.4 Laplace method | p. 189 |
10.5 General straight line | p. 190 |
10.6 LAD method in a general case | p. 191 |
11 Probability in mathematics | p. 195 |
11.1 Quadratic equations | p. 195 |
11.2 Sum and product of random numbers | p. 198 |
11.3 Socks and number theory | p. 201 |
11.4 Tshebyshev problem | p. 203 |
11.5 Random triangle | p. 205 |
11.6 Lattice-point triangles | p. 209 |
12 Matrix games | p. 211 |
12.1 Linear programming | p. 211 |
12.2 Pure strategies | p. 213 |
12.3 Mixed strategies | p. 215 |
12.4 Solution of matrix games | p. 216 |
12.5 Solution of 2 [times] 2 games | p. 218 |
12.6 Two-finger mora | p. 219 |
12.7 Three-finger mora | p. 220 |
12.8 Problem of colonel Blotto | p. 220 |
12.9 Scissors--paper--stone | p. 221 |
12.10 Birthday | p. 222 |
References | p. 223 |
Topic Index | p. 231 |
Author Index | p. 234 |