The Art Of Probability

Preface
Table of Contents
References

Chapter 1 Probability
1.1 Introduction
1.2 Models in General
1.3 The Frequency Approach Rejected
1.4 The Single Event Model
1.5 Symmetry as the Measure of Probability
1.5–1 Selected faces of a die
1.5–2 A card with the face 7
1.5–3 Probability of a spade
1.6 Independence
1.6–1 The sample space of two dice
1.6–2 The sample space of three coins
1.6–3 The number of paths
1.7 Subsets of a Sample Space
1.7–1 Exactly one head in three tosses of three coins
1.7–2 Sum of two dice Aside: Probability scales
1.8 Conditional Probability
1.8–1 At least two heads in ten tosses
1.8–2 An even sum on two dice
1.8–3 Probability of exactly three heads given that there are at least two heads
1.9 Randomness
1.9–1 The two gold coin problem
1.9–2 The two children problem
1.9–3 The four card deck
1.9–4 The two headed coin
1.9–5 No information
1.9–6 The birthday problem
1.9–7 The general case of coincidences
1.9–8 The information depends on your state of knowledge
1.10 Critique of the Model
1.A Bounds on Sums
1.B A Useful Bound

Chapter 2 Some Mathematical Tools
2.1 Introduction
2.2 Permutations
2.2–1 Arrange 3 books on a shelf
2.2–2 Arrange 6 books on a shelf
2.2–3 Arrange 10 books on a shelf
2.2–4 Two out of three items identical
2.2–5 Another case of identical items
2.3 Combinations
2.3–1 Sums of binomial coefficients
2.3–2 Bridge hands
2.3–3 3 out of 7 books
2.3–4 Probability of n heads in 2n tosses of a coin
2.3–5 Probability of a void in bridge
2.3–6 Similar items in bins
2.3–7 With at least 1 in each bin
2.4 The Binomial Distribution—Bernoulli Trials
2.4–1 Inspection of parts
2.4–2 Continued
2.4–3 Floppy discs
2.4–4 Distribution of the sum of three dice
2.4–5 Bose–Einstein statistics
2.4–6 Fermi–Dirac statistics
2.5 Random Variables, Mean and the Expected Value
2.5–1 Expected value of a roll of a die
2.5–2 Expected value of a coin toss
2.5–3 Sum of two dice
2.5–4 Gambling on an unknown bias Mathematical aside: Sums of powers of the integers
2.6 The Variance
2.6–1 Variance of a die
2.6–2 Variance of Μ equally likely outcomes
2.6–3 Variance of the sum of n dice
2.7 The Generating Function
2.7–1 Mean and variance of the binomial distribution
2.7–2 The binomial distribution again
2.7–3 The distribution of the sum of three dice
2.8 The Weak Law of Large Numbers
2.9 The Statistical Assignment of Probability
2.9–1 Test of coin and Monte Carlo methods
2.9–2 Chevalier de Mere problem
2.10 The Representation of information
2.11 Summary
2.A Derivation of the Weak Law of Large Numbers
2.B Useful Binomial Identities

Chapter 3 Methods for Solving Problems
3.1 The Five Methods
3.1–1 The elevator problem (five ways)
3.2 The Total Sample Space and Fair Games
3.2–1 Matching pennies (coins)
3.2–2 Biased coins
3.2–3 Raffles and lotteries
3.2–4 How many tickets to buy in a raffle
3.3 Enumeration
3.3–1 The game of six dice
3.3–2 The sum of three dice
3.4 Historical Approach
3.4–1 The problem of points
3.4–2 Three point game with bias
3.4–3 Estimating the bias
3.5 Recursive Approach
3.5–1 Permutations
3.5–2 Runs of heads
3.5–3 Ν biased coins
3.5–4 Random distribution of chips
3.5–5 The gambler’s ruin
3.6 The Method of Random Variables
3.6–1 Pairs of socks
3.6–2 Problem de recontre
3.6–3 Selecting Ν items from a set of Ν
3.7 Critique of the Notion of a Fair Game
3.8 Bernoulli Evaluation Mathematical aside: Log Expansions
3.8–1 Coin toss
3.8–2 Symmetric payoff
3.8–3 Fair games
3.8–4 Insurance (ideal—no extra costs)
3.9 Robustness
3.9–1 The robust birthday problem
3.9–2 The robust elevator problem
3.9–3 A variant on the birthday problem
3.9–4 A simulation of the variant birthday problem
3.10 Inclusion–Exclusion Principle
3.10–1 Misprints
3.10–2 Animal populations
3.10–3 Robust multiple sampling
3.10–4 Divisibility of numbers
3.11 Summary

Chapter 4 Countably Infinite Sample Spaces
4.1 Introduction Mathematical aside: Infinite Sums
4.2 Bernoulli Trials
4.2–1 First occurence
4.2–2 Monte Carlo test of a binomial choice
4.2–3 All six faces of a die
4.2–4 Monte Carlo estimate
4.2–5 Card collecting
4.2–6 Invariance principle of geometric distributions
4.2–7 First and second failures
4.2–8 Number of boys in a family
4.2–9 Probability that event A precedes event Β
4.2–10 Craps
4.3 On the Strategy to be Adopted
4.4 State Diagrams
4.4–1 Two heads in a row
4.4–2 Three heads in a row
4.5 Generating Functions of State Diagrams
4.5–1 Six faces of a die again
4.6 Expanding a Rational Generating Function
4.7 Checking the Solution
4.7–1 Three heads in a row
4.8 Paradoxes
4.8–1 The St. Petersburg paradox
4.8–2 The Castle Point paradox
4.8–3 Urn with black and white balls
4.9 Summary
4.A Linear Difference Equations

Chapter 5 Continuous Sample Spaces
5.1 A Philosophy of the Real Number System
5.2 Some First Examples
5.2–1 Distance from a river
5.2–2 Second random choice of angle
5.2–3 Broken stick
5.2–4 The Buffon needle
5.2–5 Another Monte Carlo estimate of π
5.2–6 Mean and variance of the unit interval
5.3 Some Paradoxes
5.3–1 Bertrand’s paradox
5.3–2 Obtuse random triangles
5.3–3 A random game
5.4 The Normal Distribution
5.4–1 Hershel’s derivation of the normal distribution
5.4–2 Distance to a random point
5.4–3 Two random samples from a normal distribution
5.5 The Distribution of Numbers
5.5–1 The general product
5.5–2 Persistence of the reciprocal distribution
5.5–3 Probability of shifting
5.5–4 The general quotient
5.6 Convergence to the reciprocal distribution
5.6–1 Product of two numbers from a flat distribution
5.6–2 Approach to the reciprocal distribution in general
5.6–3 Approach from the flat distribution
5.6–4 A computation of products from a flat distribution
5.7 Random Times
5.7–1 Sinusiodal notion
5.7–2 Random events in time
5.7–3 Mixtures
5.8 Dead Times
5.9 Poisson Distributions in time
5.9–1 Shape of the state distributions
5.10 Queuing Theory
5.11 Birth and Death Systems
5.12 Summary

Chapter 6 Uniform Probability Assignments
6.1 Mechanical Probability
6.2 Experimental Results
6.3 Mathematical Probability
6.4 Logical Probability
6.5 Robustness
6.6 Natural Variables
6.7 Summary

Chapter 7 Maximum Entropy
7.1 What is Entropy?
7.2 Shannon’s Entropy
7.2–1 An entropy computation
7.2–2 The entropy of uniform distributions
7.3 Some Mathematical Properties of the Entropy Function
7.3–1 The log inequality
7.3–2 Gibbs’ inequality
7.3–3 The entropy of independent random variables
7.3–4 The entropy decreases when you combine items
7.4 Some Simple Applications
7.4–1 Maximum entropy distributions
7.4–2 The entropy of a single binary choice
7.4–3 The entropy of repeated binary trials
7.4–4 The entropy of the time to first failure
7.4–5 A discrete distribution with infinite entropy
7.5 The Maximum Entropy Principle
7.5–1 Marginal distributions
7.5–2 Jaynes’ example
7.5–3 Extensions of Jaynes’ example
7.6 Summary

Chapter 8 Models of Probability
8.1 General Remarks
8.2 Review
8.3 Maximum Likelihood
8.3–1 Maximum likelihood in a binary choice
8.3–2 Least squares
8.3–3 Scale free
8.4 von Mises’ Probability
8.5 The Mathematical Approach
8.6 The Statistical Approach
8.7 When the Mean Does Not Exist
8.7–1 The Cauchy distribution
8.8 Probability as an Extension of Logic
8.8–1 The four liars
8.9 di Finetti
8.10 Subjective Probability
8.11 Fuzzy Probability
8.12 Probability in Science
8.13 Complex Probability
8.14 Summary

Chapter 9 Some Limit Theorems
9.1 Introduction
9.2 The Normal Distribution
9.2–1 The approximation of a unimodal distribution
9.3 The Binomial Approximation for the Case ρ = 1/2
9.3–1 Binomial sums
9.3–2 Binomial sums for nonsymmetric ranges
9.4 Approximation by the Normal Distribution
9.4–1 A normal approximation to a skewed distribution
9.4–2 Approximation to a binomial distribution
9.5 Another Derivation of the Normal Distribution
9.6 Random Times
9.7 The Zipf Distribution
9.8 Summary

Chapter 10 An Essay On Simulation
10.1 Introduction
10.2 Simulations for checking purposes
10.3 When you cannot compute the result
10.3–1 Random triangles in a circle
10.4 Direct simulations
10.5 The use of some modeling
10.6 Thought simulations
10.7 Monte Carlo methods
10.8 Some simple distributions
10.8–1 The exponential distribution
10.8–2 The normal distribution
10.8–3 The reciprocal distribution
10.9 Notes on programming many simulations
10.10 Summary

Index