The present book is based on the view that cognitive skills are best acquired by solving challenging, non-standard probability problems. The author's own experience, both in learning and in teaching, is that challenging problems often provide more, and longer lasting, inductive insights than plain-style deductions from general concepts. Problems help to develop, and to sharpen our intuition for important probabilistical concepts and tools such as conditionaing or first-step analyses.Many puzzles and problems presented here are either new within a problem solving context (although as topics in fundamental research they are of course long known) or are variations of classical problems which follow directly from elementary concepts. A small number of particularly instructive problems is taken from previous sources which in this case are generally given.
Author(s): Wolfgang Schwarz
Series: Problem books in mathematics
Edition: 1
Publisher: Springer
Year: 2008
Language: English
Pages: 134
City: New York; London
Tags: Математика;Теория вероятностей и математическая статистика;
front-matter......Page 1
Notation and Terminology......Page 12
A Tournament Problem......Page 14
How Often Do Head and Tail Occur Equally Often?......Page 15
Random Areas......Page 16
The Optimal Level of Supply......Page 17
Mixing RVs vs. Mixing Their Distributions......Page 18
Random Ranks......Page 19
Is 2X the Same as X1 + X2?......Page 20
Large Gaps......Page 21
ML Estimation with the Geometric Distribution......Page 22
The Lady Tasting Tea......Page 23
Approximately How Tall Is the Tallest?......Page 24
The Median in Samples of Exponential RVs......Page 25
Paradoxical Contribution......Page 26
Discrete Variables with Continuous Error......Page 27
The High-Resolution and the Black-White View......Page 28
The arcsin(p) Transform......Page 29
The Delta Technique with One Variable......Page 30
How Many Trials Produced a Given Maximum?......Page 31
Waiting for Success......Page 32
Mean Waiting Time for 1-1 vs. 1-2......Page 33
Random Areas......Page 34
Mixing RVs vs. Mixing Their Distributions......Page 35
Ups and Downs......Page 36
Random Powers of Random Variables......Page 37
How Many Twins Are Homozygotic?......Page 38
Approximately How Tall Is the Tallest?......Page 39
Breaking the Record......Page 40
The High-Resolution and the Black-White View......Page 41
Binomial Trials Depending on a Latent Variable......Page 42
The Delta Technique with Two Variables......Page 43
Waiting for Success......Page 44
To Begin or Not to Begin?......Page 45
A Tournament Problem......Page 46
Mean Waiting Time for 1-1 vs. 1-2......Page 47
How to Divide up Gains in Interrupted Games......Page 50
How Often Do Head and Tail Occur Equally Often?......Page 52
Sample Size vs. Signal Strength......Page 55
Birthday Holidays......Page 57
Random Areas......Page 59
Maximize Your Gain......Page 60
Maximize Your Gain When Losses Are Possible......Page 62
The Optimal Level of Supply......Page 64
Mixing RVs vs. Mixing Their Distributions......Page 66
Throwing the Same vs. Different Dice......Page 69
Random Ranks......Page 71
Ups and Downs......Page 72
Is 2X the Same as X1 + X2?......Page 73
How Many Donors Needed?......Page 74
Large Gaps......Page 76
Small Gaps......Page 77
Random Powers of Random Variables......Page 78
How Many Bugs Are Left?......Page 80
ML Estimation with the Geometric Distribution......Page 82
How Many Twins Are Homozygotic?......Page 85
The Lady Tasting Tea......Page 87
How to Aggregate Significance Levels......Page 90
Approximately How Tall Is the Tallest?......Page 92
The Range in Samples of Exponential RVs......Page 94
The Median in Samples of Exponential RVs......Page 96
Breaking the Record......Page 97
Paradoxical Contribution......Page 100
Attracting Mediocrity......Page 101
Discrete Variables with Continuous Error......Page 106
The High-Resolution and the Black-White View......Page 108
The Bivariate Lognormal......Page 113
The arcsin(p) Transform......Page 116
Binomial Trials Depending on a Latent Variable......Page 118
The Delta Technique with One Variable......Page 120
The Delta Technique with Two Variables......Page 122
How Many Trials Produced a Given Maximum?......Page 125
Waiting for Success......Page 128
References......Page 130
Index......Page 132
