# Discover Probability: How To Use It, How To Avoid Misusing It, And How It Affects Every Aspect Of Your Life

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ISBN-13: 9789814616324 World Scientific Publishing Company, Incorporated 11/14/2014 340 5.90(w) x 8.90(h) x 0.40(d)

Preface vii

Acknowledgements xiii

Session 1 What is Probability? 1

1.1 Children's Perception of Probability 3

1.2 Casting Lots in the Present and in the Past 17

1.3 Personal Interpretations of Probability 19

1.3.1 How would you choose between two equal-probability alternatives? 19

1.4 Perception of Rare Events 22

1.5 My Father's Perception of "Probability" 24

1.6 Conclusion 25

Session 2 How Do We Calculate Probabilities? 27

2.1 The Classical "Definition" of Probability 27

2.2 The Relative Frequency "Definition" 44

2.3 Conclusion 50

Session 3 The Axiomatic Approach to Probability 59

3.1 The Probability Space: (Ω, L, P) 59

3.2 The Sample Space: Ω 60

3.3 The Set of All Possible Events: L 64

3.4 The Probability Function: P 75

3.5 Summary of What we have Learned in this session 85

Session 4 Independence and Dependence Between Events 87

4.1 Correlation between Events 96

4.2 Conditional Probability 104

4.3.1 The problem 108

4.3.1.1 First solution 109

4.3.1.2 Second solution 109

4.3.1.3 Third solution 110

4.4 The General Definition of Conditional Probability 111

4.5 Correlation between Events and Conditional Probability 116

4.6 Dependence and the Extent of Overlapping between Two or More Events 124

4.7 Conditional Probability and Subjective Probability 128

4.8 Conditional Probability and Cause and Effect 130

4.8.1 Can more incriminating evidence be inculpatory? 134

4.8.2 Distinction between disjoint and independent events 138

4.9 Pairwise and Triplewise Independence between Events 139

4.9.1 Pairwise independence does not imply triplewise independence 139

4.9.2 Triplewise independence does not imply pairwise independence 140

4.9.3 Confusing conditional probability and joint probability 141

4.9.3.1 A challenging problem 144

4.9.3.2 Treasure hunt 146

4.10 Conclusion 151

Session 5 Bayes' Theorem and Its Applications 153

5.1 Newsworthy News Crazy 154

5.1.1 50% off or a rip-off? 154

5.1.2 A banana business proposition 164

5.2 False-Positive Results 167

5.3 The Three Cards Problem: Use and Misuse of Bayes' Theorem 176

5.3.1 (I) First solution 178

5.3.2 (II) Second solution 179

5.4 The Monty Hall Problem 183

5.5 The Three Prisoners' Problem: A Misuse of Probabilities That Can Cost You Your Life 185

5.5.1 The problem 186

5.5.2 The solution to the three prisoners' problem 188

5.5.2.1 First solution 188

5.5.2.2 Second solution 191

5.5.2.3 A more general but easier to solve problem 192

5.6 Re-appraisal of Probabilities 199

5.7 An Abuse of Probability and Bayes' Theorem 203

5.8 The Solution to the Urn Problem 205

5.9 Conclusion 207

Session 6 Average, Variance and Random Variables 209

6.1 To Buckle or Not to Buckle 209

6.2 Can an Average Grade be Higher Than the Highest Grade? 210

6.3 How Can One Increase the Average IQ of the Professors in Two Universities? 210

6.4 Average Speed and the Average of Two Speeds 211

6.5 Standard Deviation 217

6.6 Random Variables 221

6.7 Summary of What We Have Learned in This Session 224

Session 7 Probability Distributions 225

7.1 The Uniform Distribution 228

7.2 The Bernoulli Distribution and the Binomial Distribution 231

7.3 The Exponential Distribution 240

7.4 The Normal Distribution 245

7.5 Conclusion 248

Session 8 Shannon's Measure of Information 249

8.1 Definition of the SMI 250

8.2 Properties of the SMI 252

8.3 SMI as a Measure of Uncertainty 254

8.4 SMI as a Measure of the Amount of Information 257

8.5 The SMI of a Uniform Distribution 260

8.6 The SMI of a Non-Uniform Distribution 265

8.7 How Did Shannon Derive the SMI? 268

8.8 SMI, Entropy and the Second Law of Thermodynamics 272

Notes 279