Basic Probability

What Every Math Student Should Know

Henk Tijms

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789811202377
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Written by international award-winning probability expert Henk Tijms, Basic Probability: What Every Math Student Should Know presents the essentials of elementary probability. The book is primarily written for high school and college students learning about probability for the first time. In a highly accessible way, a modern treatment of the subject is given with emphasis on conditional probability and Bayesian probability, on striking applications of the Poisson distribution, and on the interface between probability and computer simulation.

In modern society, it is important to be able to critically evaluate statements of a probabilistic nature presented in the media in order to make informed judgments. A basic knowledge of probability theory is indispensable to logical thinking and statistical literacy. The book provides this knowledge and illustrates it with numerous everyday situations.

Contents:

• Combinatorics and Calculus for Probability
• Basics of Elementary Probability
• Useful Probability Distributions with Applications
• Surprising World of Poisson Probabilities
• Computer Simulation and Probability
• Solutions to Selected Problems

Readership: High school, college and undergraduate students exposed to probability for the first time. Elementary Probability;Poisson Distribution;Bayesian Probability;Stochastic Simulation0 Key Features:

• Unique treatment of the Bayesian probability, the Poisson distribution and the interface between probability and computer simulation
• A short book that reveals the ideas with clarity and insightfulness
• The text has 97 instructive problems with answers and fully worked-out answers for a selection of 22 problems
• The student is guided in both creative and algorithmic thinking

Chapter 1

Combinatorics and Calculus for Probability

This chapter presents a number of results from combinatorics and calculus, in preparation for the subsequent chapters. Section 1.1 introduces you to the concepts of factorials and binomial coefficients. In section 1.2 the exponential function and the natural logarithm will be discussed.

1.1 Factorials and binomial coefficients

Many probability problems require counting techniques. In particular, these techniques are extremely useful for computing probabilities in a chance experiment in which all possible outcomes are equally likely. In such experiments, one needs effective methods to count the number of outcomes in any specific event. In counting problems, it is important to know whether the order in which the elements are counted is relevant or not. Factorials and binomial coefficients will be discussed and illustrated.

In the discussion below, the fundamental principle of counting is frequently used: if there are a ways to do one activity and b ways to do another activity, then there are a × b ways of doing both. As an example, suppose that you go to a restaurant to get some breakfast. The menu says pancakes, waffles, or fried eggs, while for a drink you can choose between juice, coffee, tea, and hot chocolate. Then the total number of different choices of food and drink is 3 × 4 = 12. As another example, how many different license plates are possible when the license plate displays a nonzero digit, followed by three letters, followed by three digits? The answer is 9×26×26×26×10×10×10 = 158 184 000 license plates.

Factorials and permutations

How many different ways can you order a number of different objects such as letters or numbers? For example, what is the number of different ways that the three letters A, B, and C can be ordered? By writing out all the possibilities ABC, ACB, BAC, BCA, CAB, and CBA, you can see that the total number is 6. This brute-force method of writing down all the possibilities and counting them is naturally not practical when the number of possibilities gets large, as is the case for the number of possible orderings of the 26 letters of the alphabet. You can also determine that the three letters A, B, and C can be ordered in 6 different ways by reasoning as follows. For the first position, there are 3 available letters to choose from, for the second position there are 2 letters over to choose from, and only one letter for the third position. Therefore the total number of possibilities is 3×2×1 = 6. The general rule should now be evident. Suppose that you have n distinguishable objects. How many ordered arrangements of these objects are possible? Any ordered sequence of the objects is called a permutation. Reasoning similar to that described shows that there are n ways for choosing the first object, leaving n − 1 choices for the second object, etc. Therefore the total number of ways to order n distinguishable objects is n × (n − 1) × · · · × 2 × 1. This product is denoted by n! and is called ‘n factorial’. Thus, for any positive integer n,

A useful convention is

which simplifies the presentation of several formulas to be given below. Note that n! = n × (n − 1)! and so n! grows very quickly as n gets larger. For example, 5! = 720, 10! = 3 628 800 and 15! = 1 307 674 368 000. Summarizing, for any pos...