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 moreKnowledge 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(s): Jiri Andel
Edition: 1
Year: 2001
Language: English
Pages: 270
Mathematics of Chance......Page 5
Contents......Page 7
Preface......Page 19
Acknowledgments......Page 21
lntroduction......Page 23
1.1. Introduction......Page 27
1.2. Classical probability......Page 29
1.3. Geometric probability......Page 34
1.4. Dependence and independence......Page 36
1.5. Bayes’ theorem......Page 39
1.6. Medical diagnostics......Page 41
1.7. Random variables......Page 45
1.8. Mang Kung dice game......Page 48
1.9. Some discrete distributions......Page 50
1.10. Some continuous distributions......Page 52
2.1. Gambler’s ruin......Page 55
2.2. American roulette......Page 58
2.3. A reluctant random walk......Page 60
2.4. Random walk until no shoes are available......Page 64
2.5. Three-tower problem......Page 65
2.6. Gambler’s ruin problem with ties......Page 67
2.7. Problem of prize division......Page 69
2.8. Tennis......Page 73
2.9. Wolf and sheep......Page 76
3.1. Ticket-selling automat......Page 79
3.2. Known structure of the queue......Page 80
3.3. Queue with random structure......Page 83
3.4. Random number of customers......Page 85
4.1. Records, probability, and statistics......Page 89
4.2. Expected number of records......Page 90
4.3. Probability of r records......Page 92
4.4. Stirling numbers......Page 94
4.5. Indicators......Page 96
4.6. When records occur......Page 98
4.7. Temperature records in Prague......Page 100
4.8. How long we wait for the next record......Page 101
4.9. Some applications of records......Page 104
5.1. Geometric distribution......Page 105
5.2. Problem about keys......Page 109
5.3. Collection problems......Page 110
5.5. Waiting for a series of identical events......Page 112
5.6. Lunch......Page 113
5.7. Waiting student......Page 115
5.8. Waiting for a bus in a town......Page 116
6.1. Analysis of blood......Page 119
6.2. Overbooking airline flights......Page 121
6.3. Secretary problem......Page 123
6.5. Voting......Page 126
6.6. Dice without transitivity......Page 129
6.7. How to increase reliability......Page 131
6.8. Exam taking strategy......Page 136
6.9. Two unknown numbers......Page 139
6.11. A stacking problem......Page 141
6.12. No risk, no win......Page 143
7.1. Dormitory......Page 145
7.2. Too many marriages......Page 149
7.3. Tossing coins until all show heads......Page 150
7.4. Anglers......Page 152
7.5. Birds......Page 153
7.6. Sultan and Caliph......Page 155
7.7. Penalties......Page 156
7.8.Two 6s and two 5s......Page 157
7.9. Principle of inclusion and exclusion......Page 158
7.10. More heads on coins......Page 159
7.11. How combinatorial identities are born......Page 160
7.12. Exams......Page 161
7.13. Wyvrns......Page 162
7.14. Gaps among balls......Page 163
7.15. Numbered pegs......Page 165
7.16. Crux Mathematicorum......Page 166
7.17. Ties in elections......Page 167
7.18. Craps......Page 169
7.19. Problem of exceeding 12......Page 170
8.1. Christmas party......Page 173
8.2. Spaghetti......Page 175
8.3. Elevator......Page 177
8.4. Matching pairs of socks......Page 178
8.5. A guessing game......Page 179
8.6. Expected number of draws......Page 181
8.7. Length of the wire......Page 182
8.8. Ancient Jewish game......Page 184
8.9. Expected value of the smallest element......Page 186
8.10. Ballot count......Page 187
8.11. Bernoulli problem......Page 190
8.12. Equal numbers of heads and tails......Page 191
8.13. Pearls......Page 192
9.1. Proofreading......Page 195
9.2. How to enhance the accuracy of a measurement......Page 197
9.3. How to determine the area of a square......Page 198
9.4. Two routes to the airport......Page 201
9.5. Christmas inequality......Page 204
9.6. Cinderella......Page 206
10.1. Median......Page 211
10.2. Least squares method......Page 212
10.3. LAD method......Page 214
10.4. Laplace method......Page 215
10.5. General straight line......Page 216
10.6. LAD method in a general case......Page 217
11.1. Quadratic equations......Page 221
11.2. Sum and product of random numbers......Page 224
11.3. Socks and number theory......Page 227
11.4. Tshebyshev problem......Page 229
11.5. Random triangle......Page 231
11.6. Lattice-point triangles......Page 235
12.1. Linear programming......Page 237
12.2. Pure strategies......Page 239
12.3. Mixed strategies......Page 241
12.4. Solution of matrix games......Page 242
12.5. Solution of 2 x 2 games......Page 244
12.6. Two-finger mora......Page 245
12.8. Problem of colonel Blotto......Page 246
12.9. Scissors—paper—stone......Page 247
12.10. Birthday......Page 248
References......Page 249
Topic Index......Page 257
Author Index......Page 260
