**Introduction **

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments. This text is designed for an introductory probability course taken by sophomores, juniors, and seniors in mathematics, the physical and social sciences, engineering, and computer science. It presents a thorough treatment of probability ideas and techniques necessary for a firm understanding of the subject. The text can be used in a variety of course lengths, levels, and areas of emphasis. For use in a standard one-term course, in which both discrete and continuous probability is covered, students should have taken as a prerequisite two terms of calculus, including an introduction to multiple integrals. In order to cover Chapter 11, which contains material on Markov chains, some knowledge of matrix theory is necessary. The text can also be used in a discrete probability course. The material has been organized in such a way that the discrete and continuous probability discussions are presented in a separate, but parallel, manner. This organization dispels an overly rigorous or formal view of probability and offers some strong pedagogical value in that the discrete discussions can sometimes serve to motivate the more abstract continuous probability discussions. For use in a discrete probability course, students should have taken one term of calculus as a prerequisite.

**Contents **

**1 Discrete Probability Distributions **

1.1 Simulation of Discrete Probabilities

1.2 Discrete Probability Distributions

**2 Continuous Probability Densities **

2.1 Simulation of Continuous Probabilities

2.2 Continuous Density Functions .

** 3 Combinatorics **

3.1 Permutations

3.2 Combinations

3.3 Card Shuffling

**4 Conditional Probability 133 **

4.1 Discrete Conditional Probability

4.2 Continuous Conditional Probability

4.3 Paradoxes

**5 Distributions and Densities **

5.1 Important Distributions

5.2 Important Densities

6 Expected Value and Variance

6.1 Expected Value

6.2 Variance of Discrete Random Variables

6.3 Continuous Random Variables

**7 Sums of Random Variables **

7.1 Sums of Discrete Random Variables

7.2 Sums of Continuous Random Variables

** 8 Law of Large Numbers **

8.1 Discrete Random Variables

8.2 Continuous Random Variables

**9 Central Limit Theorem **

9.1 Bernoulli Trials

9.2 Discrete Independent Trials .

9.3 Continuous Independent Trials . .

**10 Generating Functions 365 **

10.1 Discrete Distributions .

10.2 Branching Processes

10.3 Continuous Densities

**11 Markov Chains **

11.1 Introduction

11.2 Absorbing Markov Chains

11.3 Ergodic Markov Chains

11.4 Fundamental Limit Theorem

11.5 Mean First Passage Time

**12 Random Walks**

12.1 Random Walks in Euclidean Space

12.2 Gambler’s Ruin

12.3 Arc Sine Laws .

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