Introduction to Probability, Second Edition

Synopsis

Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and toolsfor understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory.

The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces.

The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.

The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources.

Supplementary material is available on Joseph Blitzstein's website www. stat110.net. The supplements include:
Solutions to selected exercises
Additional practice problems
Handouts including review material and sample exams Animations and interactive visualizations created in connection with the edX online version of Stat 110.
Links to lecture videos available on ITunes U and YouTube There is also a complete instructor's solutions manual available to instructors who require the book for a course.

Summary

Chapter 1: Introduction

* Definition of probability and its role in decision-making.
* Applications of probability in various fields.
* Terminology and basic concepts, such as sample space, events, and outcomes.

Example: A coin is tossed. The sample space consists of {H, T} (heads, tails). The event "heads" is {H}. The probability of heads is 1/2.

Chapter 2: Basic Combinatorics

* Counting principles: addition rule, multiplication rule, factorial.
* Permutations and combinations.
* Applications in probability calculations.

Example: A committee of 5 people is to be selected from a group of 10. There are 252 different possible committees.

Chapter 3: Conditional Probability and Independence

* Conditional probability and its definition.
* Independence of events.
* Bayes' theorem and its applications.

Example: The probability of rain is 0.3. Given that it rains, the probability of thunder is 0.4. The probability of both rain and thunder is 0.12.

Chapter 4: Discrete Probability Distributions

* Introduction to discrete probability distributions.
* Common distributions: binomial, geometric, hypergeometric.
* Parameters and properties of each distribution.

Example: A fair coin is tossed 5 times. The probability of getting exactly 3 heads is given by the binomial distribution with parameters n = 5, p = 1/2.

Chapter 5: Continuous Probability Distributions

* Introduction to continuous probability distributions.
* Common distributions: uniform, normal, exponential.
* Properties and applications of each distribution.

Example: The heights of adult males are normally distributed with mean 175 cm and standard deviation 7 cm. The probability of a randomly selected male being taller than 185 cm is 0.1587.

Chapter 6: Joint Probability Distributions

* Joint probability distributions for multiple random variables.
* Marginal and conditional distributions.
* Applications in various scenarios.

Example: Two dice are rolled. The joint probability distribution gives the probability of each possible pair of outcomes. The marginal distribution for the first die shows the probability of getting each number.

Chapter 7: Expectation and Variance

* Expectation of a random variable.
* Variance and standard deviation.
* Properties and applications of these concepts.

Example: The expected number of heads when a fair coin is tossed 10 times is 5. The variance is 2.5.

Chapter 8: Limit Theorems

* Laws of large numbers and central limit theorem.
* Applications in real-world phenomena.
* Convergence of sample means to population mean.

Example: The average height of a large sample of adult males will be approximately normally distributed, even if the individual heights are not normally distributed.

Chapter 9: Bayesian Inference

* Bayes' theorem as a framework for updating probabilities.
* Applications in hypothesis testing, parameter estimation, and decision analysis.

Example: A company receives 5% defective products from their supplier. They randomly inspect a batch of 100 products and find 3 defective ones. The probability that the batch is actually defective is 0.4792.