# MA252: Introduction to Probability Theory

Unit 1: Introduction to Probability   This unit will introduce you to fundamental concepts of probability theory. You will learn the definitions of probability, random variables, outcome space, events, and probability function. These concepts of probability will be explored through simple chance experiments with discrete outcomes, such as tossing a coin or rolling a die. Random variables represent outcomes of chance experiments. You will also learn about four basic set operations - union, intersection, difference, and complement. This unit will also introduce you to the concepts of conditional probability (i.e., the probability of event A, given the occurrence of some other event B) and Bayes’Theorem, which is one of the most celebrated theorems in the theory of probability.

Completing this unit should take approximately 34.25 hours.

☐    Subunit 1.1: 7 hours

☐    Subunit 1.2: 10.5 hours

☐    Subunit 1.3: 3 hours

☐    Subunit 1.4: 5 hours

☐    Subunit 1.5: 2.5 hours

☐    Subunit 1.6: 6.25 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to: - define probability, sample space, events, and probability functions;
- use combinations to evaluate the probability of outcomes in coin-flipping experiments;
- calculate the probability of the union and intersection of events;
- define and calculate conditional probability; and
- apply Bayes’ theorem to simple situations.

1.1 Probability and Set Operations   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Jeremy Orloff and Professor Jonathan Bloom’s Math 18.05: Introduction to Probability and Statistics: “Lecture 1: Counting and Sets” and “Lecture 2: Probability: Terminology and Examples” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Jeremy Orloff and Professor Jonathan Bloom’s Math 18.05: Introduction to Probability and Statistics: “Lecture 1: Counting and Sets” (PDF) and “Lecture 2: Probability: Terminology and Examples” (PDF)

Instructions: Read both sets of lecture notes for an introduction to the basic concepts of probability. If one performs an experiment that has random outcomes, the set of all possible outcomes, S, is called the sample space, and the elements of S are called sample points. Subsets of S are called events and are usually denoted by letters like A, B, E, F, etc....

The probability of an event A, usually denoted by P(A), is defined by k/n, where n is the total number of possible outcomes of the experiment and k is the number of outcomes that result in the occurrence of A. Since events are sets, one needs to know about set operations. This lecture will give you more details about sets and how to compute probabilities of events.

Reading these lecture notes should take approximately 1 hour.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Jeremy Orloff, Jonathan Bloom, and Massachusetts Institute of Technology OpenCourseWare.

• Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 1: Introduction: Probability and Counting” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 1: Introduction: Probability and Counting” (YouTube)

Instructions: Watch this video. It will introduce you to the basic concepts of probability, including outcome space, events, and probability functions. Several examples will be given to show you how to compute probabilities of events.

Watching this video and taking notes should take approximately 1 hour and 30 minutes.

• Lecture: Khan Academy: Salman Khan’s Probability Lecture Series: “Probability (1)” and “Probability (2)” Link: Khan Academy: Salman Khan’s Probability Lecture Series: “Probability (1)” (YouTube) and “Probability (2)” (YouTube)

Instructions: Watch these videos. The first video defines probability and the second explores the outcomes from coin flips.

Watching these videos and taking notes should take approximately 30 minutes.

• Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 1, Section 1.2: Discrete Probability Distributions” Link: Dartmouth College: Professors Charles M.Grinstead and J. Laurie Snell’s Introduction to Probability“Chapter 1, Section 1.2: Discrete Probability Distributions” (PDF)

Instructions: Read pages 18 - 29 of Section 1.2 “Discrete Probability Distributions” in Chapter 1. This reading will introduce you to random variables and probability distributions. You will also learn about several important properties of probability distributions. Lastly, you will be shown how to use distribution functions to compute probabilities of events, their unions, and their intersections.

Reading this chapter and taking notes should take approximately 2 hours.

• Assessment: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 1, Section 1.2: Exercises” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 1, Section 1.2: Exercises” (PDF)

Instructions: Go to pages 35 - 40 and complete exercises 2, 4, 5, 6, 7, 9, 10, 19, and 21. You can then check your answers to odd-numbered questions here

Completing this assessment should take approximately 2 hours.

1.2 Properties of Probability and Counting Techniques   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 2: Properties of Probability” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 2: Properties of Probability” (PDF)

Instructions: This lecture will introduce you to the basic properties of probability. It will also teach you the basic principles of counting, such as the multiplication rule, permutations, and combinations.

A permutation of n objects taken r at a time is the arrangement of the n objects in a specific order using r objects at a time. It is usually denoted by nPr.

A combination of n objects taken r at a time is the number of ways we can select r objects from a set of n objects without regard to order. It is usually denoted by nCr.

Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

• Lecture: Khan Academy: Salman Khan’s Probability Lecture Series: “Probability Using Combinations” Link: Khan Academy: Salman Khan’s Probability Lecture Series: “Probability Using Combinations” (YouTube)

Instructions: Watch this video. It will help you learn how to calculate probability from coin flips by using combinations.

Watching this video and taking notes should take approximately 15 minutes.

• Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 2: Probability Functions” and “Lecture 3: Permutations” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 2: Probability Functions” (YouTube) and “Lecture 3: Permutations” (YouTube)

Instructions: Watch these videos. They will introduce you to probability functions and the counting techniques, including permutations and combinations.

Watching these videos and taking notes should take approximately 3 hours.

• Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 3, Section 3.1: Permutations” and “Chapter 3, Section 3.2: Combinations” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability“Chapter 3, Section 3.1: Permutations” (PDF) and “Chapter 3, Section 3.2: Combinations” (PDF)

Instructions: Read pages 75 - 84 of Section 3.1 “Permutations” and pages 92 - 101 of Section 3.2 “Combinations” in Chapter 3. This reading will introduce you to the two important counting techniques involving permutations and combinations.

Reading these textbook sections and taking notes should take approximately 2 hours and 45 minutes.

• Assessment: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 3, Section 3.1: Exercises” and “Chapter 3, Section 3.2: Exercises” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 3, Section 3.1: Exercises” (PDF) and “Chapter 3, Section 3.2: Exercises” (PDF)

Instructions: Go to the exercises for Section 3.1 on page 88 and page 89 and complete exercises 1, 2, 3, 5, 6, 10, 12, and 13. Then go to the exercises for Section 3.2 on pages 113 - 116 and complete exercises 2, 10, 11, 12, 19, and 20. You can then check your answers to odd-numbered questions here

Completing this assessment should take approximately 3 hours.

1.3 Union of Events   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 3: Probabilities of Unions of Events” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 3: Probabilities of Unions of Events” (PDF)

Instructions: Read this lecture, which will help you learn how to calculate the probability of unions of events. It will also review some of the counting formulas involving permutations and combinations and will discuss multinomial coefficients.

Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

• Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 4: Probability Functions (Continued)” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 4: Probability Functions (Continued)” (YouTube)

Instructions: Watch this video for an introduction to more properties of probability functions and how to use them to solve problems.

Watching this video and taking notes should take approximately 1 hour and 30 minutes.

1.4 Conditional Probability   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 4: Conditional Probability” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 4: Conditional Probability” (PDF)

Instructions: Read this lecture to learn about the definition of conditional probability. Calculations of conditional probability are illustrated via a simple example. This lecture will remind you about unions of events and how to compute their probabilities. It will define the conditional probability of an event A given another event B, denoted by P(A|B), which is the probability that A occurs given that B has already occurred. It will also show how conditional probability is used to compute the probability of intersections of events.

Reading this lecture and taking notes should take approximately 1 hour and 30 minutes.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

• Lecture: Khan Academy: Salman Khan’s Probability Lecture Series: “Conditional Probability and Combinations” Link: Khan Academy: Salman Khan’s Probability Lecture Series: “Conditional Probability and Combinations” (YouTube)

Instructions: Watch this video. It will help you learn how to calculate the conditional probability of randomly picking a fair coin out of a mixture of fair and biased coins given that the coin turned heads 4 times in 6 flips.

Watching this video and taking notes should take approximately 30 minutes.

• Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 5: Conditional Probability” and “Lecture 6: Conditional Probability (Continued)” Link: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences“Lecture 5: Conditional Probability” (YouTube) and “Lecture 6: Conditional Probability (Continued)” (YouTube)

Instructions: Watch these videos. They will introduce you to conditional probability, its properties, and how to use it to solve problems.

Watching these videos and taking notes should take approximately 3 hours.

1.5 Independent Events   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 5: Independence of Events, Bayes’ Theorem” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 5: Independence of Events, Bayes’ Theorem” (PDF)

Instructions: Read this lecture for an introduction to one of the most important formulas in probability theory - Bayes’ Theorem. This reading will demonstrate the power of Bayes’ theorem through several real-world examples. You will first learn that 2 events A and B are independent if P(A and B) = P(A)P(B), which means that the occurrence of one of the two events, for instance A, does not affect the occurrence of the other event, B. Another interpretation using conditional probability will be given as well, and, finally, an important law used in Bayes’ formula, called the law of total probability, will be given.

Reading this lecture and taking notes should take approximately 1 hour.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

• Lecture: YouTube: UCLA: Professor Herbert Enderton’s Math 3C: Math and Probability for Life Sciences: “Lecture 7: Independent Events” Link: YouTube: UCLA: Professor David Welsbart’s Math 3C: Math and Probability for Life Sciences“Lecture 7: Independent Events” (YouTube)

Instructions: Watch this video for an introduction to independent events.

Watching this video and taking notes should take approximately 1 hour and 30 minutes.

1.6 Bayes’ Theorem   - Reading: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics: “Lecture 7: Bayes’ Formula” Link: Massachusetts Institute of Technology OpenCourseWare: Professor Dmitry Panchenko’s Math 18.05: Introduction to Probability and Statistics“Lecture 7: Bayes’ Formula” (PDF)

Instructions: Read this lecture, which is a continuation of Lecture 6 from the previous subunit. The lecture starts with the law of total probability and then shows how Bayes’ formula is just a consequence of that law combined with the definition of conditional probability. Bayes’ formula allows us to find the probability of the first stage of an experiment given that we know the outcome of the second stage.

Reading this lecture and taking notes should take approximately 1 hour.

Terms of Use: The above material is released under a Creative Commons Attribution-Non Commercial-ShareAlike License 3.0. It is attributed to Dmitry Panchenko and Massachusetts Institute of Technology OpenCourseWare.

Instructions: Watch this video, which gives another overview of Bayes’ Theorem.

Watching this video and taking notes should take approximately 15 minutes.

• Reading: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 4, Section 4.1: Discrete Conditional Probability” Link: Dartmouth College: Professors Charles M. Grinstead and J. Laurie Snell’s Introduction to Probability: “Chapter 4, Section 4.1: Discrete Conditional Probability” (PDF)

Instructions: Read pages 133 - 147 of Section 4.1 “Discrete Conditional Probability” in Chapter 4. This reading will introduce you to conditional probability, which you saw in Subunit 1.4, then explain Bayes’ Rule and talks about independence of events.

Reading this chapter and taking notes should take approximately 2 hours and 30 minutes.