Can we correctly predict the flip of a fair coin more than half the time — or the decay of a single radioactive atom? Our intuition, based on a lifetime of experience, tells us that we cannot, as these are classic examples of what are known to be 50–50 guesses.
But mathematics is filled with counterintuitive results — and this book discusses some surprising and entertaining examples. It is possible to devise experiments in which a flipped coin lands heads completely at random half the time, but we can also correctly predict when it will land heads more than half the time. The Fate of Schrodinger's Cat shows how high-school algebra and basic probability theory, with the invaluable assistance of computer simulations, can be used to investigate both the intuitive and the counterintuitive.
This book explores fascinating and controversial questions involving prediction, decision-making, and statistical analysis in a number of diverse areas, ranging from whether there is such a thing as a "hot hand" in shooting a basketball, to how we can successfully predict, more than half the time, the decay of the radioactive atom that determines the fate of Schrodinger's Cat.
Author(s): James D Stein
Series: Problem Solving in Mathematics and Beyond
Edition: 1
Publisher: WSPC
Year: 2020
Language: English
Pages: 172
Tags: Mathematics; Probability; Simulation; Statistical Analysis; Monty Hall Problem; Blackwell's Bet
Contents
Preface
Introduction — Mathematics, Intuition, and Computers
Section I: The Realm of the Counterintuitive
Chapter 1 The Monty Hall Problem
1.1 The Monty Hall Problem
1.2 Looking at an Extreme Case
1.3 A Trickier Monty Hall Problem
1.4 Just One Look
Chapter 2 How Probabilistic Entanglement Connects Almost Everything
2.1 Is Everything Connected?
2.2 Quantum Entanglement
2.3 Probabilistic Entanglement
2.4 Benefitting from a Coin Flip
2.5 Flipism
2.6 Multiple Observations from a Single Group
2.7 An Odd Number of Trials
2.8 ComparingMeans of Unrelated Groups
2.9 How Fundamental Is Probability?
Chapter 3 Blackwell’s Bet
3.1 Wunch with Wenny
3.2 Unexpected Expectations
3.3 Can Blackwell’s Bet Help You Beat the Line at Sports Betting?
3.4 Applying Blackwell’s Bet to Sample and Population Statistics
Chapter 4 A Stop at Willoughby —Mathematics in the Twilight Zone
4.1 Can You Predict the Flip of a Coin?
4.2 RandomWalks
4.3 Next Stop— Willoughby
4.4 An Actual Coin Flip Prediction
4.5 A Magical Mystery Tour
4.6 AreWe Predicting the Future?
Chapter 5 The Fate of Schrodinger’s Cat
5.1 Blackwell’s Bet Redux
5.2 More About Bernoulli Trials
5.3 Creating a Predictable Schrodinger’s Cat Experiment
5.4 Would This Experiment Fool Erwin Schrodinger?
5.5 The Schrodinger Switcheroo
5.6 Why Science Is Difficult
5.7 The Solar Neutrino Deficit
5.8 Hidden Variables
5.9 Non-Predictable Bernoulli Trials
5.10 Time Travel and Predictability Paradoxes
5.11 Of Time and Third Avenue
5.12 Checking Out the Schrodinger’s Cat Experiment in Your Home
Chapter 6 Coins and Camels
6.1 Distinguishing Similar Bernoulli Trials
6.2 The Problem of the 17 Camels
6.3 Doing the Math
6.4 A Slightly Different Problem
6.5 When One Door Closes—How Mathematicians Find Problems to Investigate
Section II: The Monday Morning Quarterback
Chapter 7 The Joy of Simulation
7.1 Random Number Generators
7.2 Chi-Square Tests; Karl Pearson
7.3 Simulations in Contemporary Science and Engineering
7.4 Simulation in Fantasy Sports
7.5 Why Educators Should Teach Simulation Ratherthan Algebra
7.6 Simulation in the Electoral College
7.7 Why are Tennis’ Big Three so Dominant?
Chapter 8 Numbed by Numbers
8.1 A Really, Really, REALLY Bad Statistic
8.2 The Year of the Unbeatens
8.3 Simulating the NFL
8.4 Integrating the RealWorld with Education
Chapter 9 Losing the Battle, Winning the War
9.1 The Post-Season
9.2 The Gibbard-Satterthwaite Theorem
9.3 Dumping for Future Advantage
Section III: Getting It Right; A Synergy of Mathematics, Intuition and Computers
Chapter 10 The Hot Hand
10.1 Can the “Hot Hand” Be Exploited to Win at Betting Sports?
10.2 The “Hot Hand” in aWider Context
Chapter 11 The Bent Coin and the Hot Hand
11.1 Binomial Trials and Tribulations
11.2 Binomial Trials and Flipping a Coin Just Once
11.3 A Different Approach
11.4 The Probability of the Probability of Probabilities
11.5 More Simulations
11.6 The Hot Hand Redux
11.7 Jensen’s Inequality
11.8 Doing Better than Average
Section IV: The Last Hurrah
Chapter 12 Using Combinatorics to Improve Advertising — For Everyone
12.1 A Brief History of Advertising
12.2 WhatWe Hate about Advertising
12.3 The Element of Surprise
12.4 Combinatorial Commercials
Appendix —Basic Probability Theory
A.1 Computational Rules for Probability
A.2 Conditional Probability
A.3 Independent Events
A.4 Expected Value (a.k.a. Expectation)
A.5 Bernoulli Trials
A.6 Means and Medians
Annotated Bibliography
Index
