40 puzzles and problems in probability and mathematical statistics

As a student I discovered in our library a thin booklet by Frederick Mosteller entitled50 Challenging Problems in Probability. Itreferredtoas- plementary "regular textbook" by William Feller, An Introduction to Pro- bilityTheoryanditsApplications.SoItookthisonealong,too,andstartedon the ?rst of Mosteller's problems on the train riding home. From that evening, I caught on to probability. These two books were not primarily about abstract formalisms but rather about basic modeling ideas and about ways -- often extremely elegant ones -- to apply those notions to a surprising variety of empirical phenomena. Essentially, these books taught the reader the skill to "think probabilistically" and to apply simple probability models to real-world problems. The present book is in this tradition; it is based on the view that those cognitive skills are best acquired by solving challenging, nonstandard pro- bility problems. My own experience, both in learning and in teaching, is that challenging problems often help to develop, and to sharpen, our probabilistic intuition much better than plain-style deductions from abstract concepts.

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Choice Review

In this work, Schwarz (psychology, Univ. of Potsdam, Germany) provides a collection of problems involving probability theory and mathematical statistics. The book is divided into three sections. In the first part, the author states 40 problems. The second section contains hints for each of the problems, with complete solutions provided in the third section. Schwarz assumes that readers are familiar with probability theory, calculus, and some aspects of mathematical statistics. His goal is to help students develop their probabilistic intuition and problem-solving skills by solving challenging problems that require careful reasoning rather than difficult computations. This book may be of interest to advanced students studying probability theory. It is very similar in spirit to Frederick Mosteller's Fifty Challenging Problems in Probability with Solutions (1965), although the problems presented here require somewhat more advanced mathematics. Summing Up: Recommended. Libraries serving upper-division undergraduates and graduate students. B. Borchers New Mexico Institute of Mining and Technology

0 Notation and Terminology	p. 1
1 Problems	p. 3
1.1 To Begin or Not to Begin?	p. 3
1.2 A Tournament Problem	p. 3
1.3 Mean Waiting Time for 1 - 1 vs. 1 - 2	p. 4
1.4 How to Divide up Gains in Interrupted Games	p. 4
1.5 How Often Do Head and Tail Occur Equally Often?	p. 4
1.6 Sample Size vs. Signal Strength	p. 5
1.7 Birthday Holidays	p. 5
1.8 Random Areas	p. 5
1.9 Maximize Your Gain	p. 6
1.10 Maximize Your Gain When Losses Are Possible	p. 6
1.11 The Optimal Level of Supply	p. 6
1.12 Mixing RVs vs. Mixing Their Distributions	p. 7
1.13 Throwing the Same vs. Different Dice	p. 8
1.14 Random Ranks	p. 8
1.15 Ups and Downs	p. 9
1.16 Is 2X the Same as X[subscript 1] + X[subscript 2]?	p. 9
1.17 How Many Donors Needed?	p. 10
1.18 Large Gaps	p. 10
1.19 Small Gaps	p. 11
1.20 Random Powers of Random Variables	p. 11
1.21 How Many Bugs Are Left?	p. 11
1.22 ML Estimation with the Geometric Distribution	p. 11
1.23 How Many Twins Are Homozygotic?	p. 12
1.24 The Lady Tasting Tea	p. 12
1.25 How to Aggregate Significance Levels	p. 13
1.26 Approximately How Tall Is the Tallest?	p. 13
1.27 The Range in Samples of Exponential RVs	p. 14
1.28 The Median in Samples of Exponential RVs	p. 14
1.29 Breaking the Record	p. 15
1.30 Paradoxical Contribution	p. 15
1.31 Attracting Mediocrity	p. 16
1.32 Discrete Variables with Continuous Error	p. 16
1.33 The High-Resolution and the Black-White View	p. 17
1.34 The Bivariate Lognormal	p. 18
1.35 The arcsin ([radical]p) Transform	p. 18
1.36 Binomial Trials Depending on a Latent Variable	p. 19
1.37 The Delta Technique with One Variable	p. 19
1.38 The Delta Technique with Two Variables	p. 20
1.39 How Many Trials Produced a Given Maximum?	p. 20
1.40 Waiting for Success	p. 21
2 Hints	p. 23
2.1 To Begin or Not to Begin?	p. 23
2.2 A Tournament Problem	p. 23
2.3 Mean Waiting Time for 1 - 1 vs. 1 - 2	p. 23
2.4 How to Divide up Gains in Interrupted Games	p. 24
2.5 How Often Do Head and Tail Occur Equally Often?	p. 24
2.6 Sample Size vs. Signal Strength	p. 24
2.7 Birthday Holidays	p. 24
2.8 Random Areas	p. 24
2.9 Maximize Your Gain	p. 25
2.10 Maximize Your Gain When Losses Are Possible	p. 25
2.11 The Optimal Level of Supply	p. 25
2.12 Mixing RVs vs. Mixing Their Distributions	p. 25
2.13 Throwing the Same vs. Different Dice	p. 26
2.14 Random Ranks	p. 26
2.15 Ups and Downs	p. 26
2.16 Is 2X the Same as X[subscript 1] + X[subscript 2]?	p. 27
2.17 How Many Donors Needed?	p. 27
2.18 Large Gaps	p. 27
2.19 Small Gaps	p. 27
2.20 Random Powers of Random Variables	p. 27
2.21 How Many Bugs Are Left?	p. 28
2.22 ML Estimation with the Geometric Distribution	p. 28
2.23 How Many Twins Are Homozygotic?	p. 28
2.24 The Lady Tasting Tea	p. 29
2.25 How to Aggregate Significance Levels	p. 29
2.26 Approximately How Tall Is the Tallest?	p. 29
2.27 The Range in Samples of Exponential RVs	p. 30
2.28 The Median in Samples of Exponential RVs	p. 30
2.29 Breaking the Record	p. 30
2.30 Paradoxical Contribution	p. 31
2.31 Attracting Mediocrity	p. 31
2.32 Discrete Variables with Continuous Error	p. 31
2.33 The High-Resolution and the Black-White View	p. 31
2.34 The Bivariate Lognormal	p. 32
2.35 The arcsin ([radical]p) Transform	p. 32
2.36 Binomial Trials Depending on a Latent Variable	p. 32
2.37 The Delta Technique with One Variable	p. 33
2.38 The Delta Technique with Two Variables	p. 33
2.39 How Many Trials Produced a Given Maximum?	p. 34
2.40 Waiting for Success	p. 34
3 Solutions	p. 35
3.1 To Begin or Not to Begin?	p. 35
3.2 A Tournament Problem	p. 36
3.3 Mean Waiting Time for 1 - 1 vs. 1 - 2	p. 37
3.4 How to Divide up Gains in Interrupted Games	p. 40
3.5 How Often Do Head and Tail Occur Equally Often?	p. 42
3.6 Sample Size vs. Signal Strength	p. 45
3.7 Birthday Holidays	p. 47
3.8 Random Areas	p. 49
3.9 Maximize Your Gain	p. 50
3.10 Maximize Your Gain When Losses Are Possible	p. 52
3.11 The Optimal Level of Supply	p. 54
3.12 Mixing RVs vs. Mixing Their Distributions	p. 56
3.13 Throwing the Same vs. Different Dice	p. 59
3.14 Random Ranks	p. 61
3.15 Ups and Downs	p. 62
3.16 Is 2X the Same as X[subscript 1] + X[subscript 2]?	p. 63
3.17 How Many Donors Needed?	p. 64
3.18 Large Gaps	p. 66
3.19 Small Gaps	p. 67
3.20 Random Powers of Random Variables	p. 68
3.21 How Many Bugs Are Left?	p. 70
3.22 ML Estimation with the Geometric Distribution	p. 72
3.23 How Many Twins Are Homozygotic?	p. 75
3.24 The Lady Tasting Tea	p. 77
3.25 How to Aggregate Significance Levels	p. 80
3.26 Approximately How Tall Is the Tallest?	p. 82
3.27 The Range in Samples of Exponential RVs	p. 84
3.28 The Median in Samples of Exponential RVs	p. 86
3.29 Breaking the Record	p. 87
3.30 Paradoxical Contribution	p. 90
3.31 Attracting Mediocrity	p. 91
3.32 Discrete Variables with Continuous Error	p. 96
3.33 The High-Resolution and the Black-White View	p. 98
3.34 The Bivariate Lognormal	p. 103
3.35 The arcsin ([radical]p) Transform	p. 106
3.36 Binomial Trials Depending on a Latent Variable	p. 108
3.37 The Delta Technique with One Variable	p. 110
3.38 The Delta Technique with Two Variables	p. 112
3.39 How Many Trials Produced a Given Maximum?	p. 115
3.40 Waiting for Success	p. 118
References	p. 121
Index	p. 123

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