STAT 325 Introduction to Probability Models Spring 2010

Course Syllabus

Instructor: Allan Rossman
Class Times: MTuWTh 12:10-1:00, room 186-C102
Office: Faculty Office Building East 25-102
Email: arossman@calpoly.edu
Office Hours: MW 9:30-10:30, TuTh 1:30-3:00, and by appointment and by chance
Text: Concepts in Probability and Stochastic Modeling, by James J. Higgins and Sallie Keller-McNulty

Course Webpage: http://statweb.calpoly.edu/arossman/stat325/

This website contains much information, including the course syllabus, handouts, assignments, solutions, and announcements.

Overview:

“We see that the theory of probability is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it....It is remarkable that this science, which originated in the consideration of games of chance, should have become the most important object of human knowledge....The most important questions of life are, for the most part, really only problems of probability.” – Pierre Simon, Marquis de Laplace

“The greatest challenge in understanding the role of randomness in life is that although the basic principles of randomness arise from everyday logic, many of the consequences that follow from those principles prove counterintuitive.” – Leonard Mlodinow, The Drunkard’s Walk: How Randomness Rules Our Lives

Probability is the mathematical study of uncertainty or randomness. Although the famous mathematician Laplace no doubt overstated the point in the passage above, there is no denying that the concept of uncertainty pervades our everyday lives and that the mathematics of probability has become a very useful tool in a wide variety of fields.  But as Dr. Mlodinow’s remarks, many studies have shown that human intuition does not generally deal well with issues of uncertainty. This speaks to why it’s important to undertake a careful and systematic study of randomness. This course provides an introduction to the fundamental concepts and methods of probability with an emphasis on their applications to many disciplines as well as everyday life.

Preview: To whet your appetite for the types of questions that we will investigate, I present this sampling:

1. If a hospital were to distribute a group of newborn babies to their mothers at random, how many mothers would we expect to get paired with the correct baby?
2. Based on forensic evidence obtained at the scenes of the crimes, how did probability arguments help a jury to reach a verdict in the celebrated case of “Sneaky,” the man accused of raping several women in the Shadyside district of Pittsburgh in 1985-86?
3. How likely is it that you and a friend will meet for lunch, if your arrival times are random and you agree to wait only for a certain length of time?
4. If a person tests positive for the AIDS virus, what is the probability that you have actually been infected?
5. Why can a fickle contestant double his/her chances of winning the jackpot on a game show?
6. Can the rules for a series of sports games be modified in order to identify the better team with fewer games played?
7. How many people must be in a room in order to have a 70% chance that at least two of them share the same birthday?
8. What is the probability that you would pass a multiple choice exam by sheer guessing?
9. A pregnant woman writes to Dear Abby saying that her military husband is the father of her child even though it was 310 days between her husband’s last visit and her child's birth! What are the chances of that?
10. Would the average waiting time of customers be shorter if fast food restaurant uses a single waiting line or multiple lines?
11. How can the Army determine which soldiers have a certain disease without performing the expensive blood test needed to detect the disease on each and every soldier?
12. A much higher percentage of men than of women was admitted to the graduate school of the University of California at Berkeley in the fall quarter of 1973. Can we attribute the difference to sex discrimination?

Prerequisites: The prerequisites for this course are knowledge of differential and integral calculus for one variable, covered by MATH 141, 142, 143; basic ideas of linear algebra, covered by MATH 206; and fundamentals of computer programming, covered by any introductory computer science course. We will also make occasional use of set-theoretic properties and operations, which we will review briefly at the appropriate time. No prior knowledge of or familiarity with probability or statistics is assumed or expected.

Goals: By the conclusion of the course, I hope that you have acquired the ability to:

• understand fundamental concepts of the mathematical theory of probability, including conditional probability, independence, random variables, probability distributions, and expectations;
• perform a wide variety of probability calculations and derivations;
• apply ideas and techniques associated with Markov chains, Poisson process, and reliability models to solve problems;
• use computer software as a tool for solving probability problems;
• write computer programs to simulate and analyze random processes;
• approach and solve practical problems through probabilistic reasoning.

• communicate your knowledge of probability effectively.

Course materials: You should obtain the textbook and a three-ring binder for organizing your notes.  I will post handouts for every class session by the previous evening. I think you’ll find it helpful to print these and bring them to class with you. You will find much helpful information posted at the course website, so please check there often. You must also have a calculator and access to the two statistical software packages described below. 

Classroom culture: Please come to class prepared and willing (preferably eager!) to engage in class and answer my questions and collaborate with your peers and ask questions of me.  This will not only help you to learn the material and perform well in the course, but it will also produce a much more enjoyable learning environment for all of us.  Class attendance is very strongly encouraged, as I hope that our class sessions will prove to be valuable learning experiences.  Needless to say, you are responsible for everything presented in class.

I also expect you to devote substantial outside-of-class time to your work for this course, typically involving 8-12 hours per week.  I anticipate that this work will be divided among:

• reviewing your class notes
• reading the textbook
• solving homework problems
• working on investigation assignments
• preparing for exams

Computer use: We will make extensive use of computers in this course. They will prove useful in at least three ways:

1. for facilitating probability calculations;

• Many of the calculations that we’ll learn are very cumbersome to do by hand.

2. for addressing “what if” questions that allow you to explore probability concepts;

• We can investigate the effects of tweaking various aspects of problems by obtaining immediate feedback.

3. for conducting simulations to investigate long-run behavior of random phenomena;

• Simulation is an extremely powerful tool, not only for exploring probability concepts but also for solving problems that are too complex to be analyzed analytically.

For these purposes we will make frequent use of two statistics package: Minitab and R.  No prior knowledge of these software tools is assumed. Minitab is a widely used, user-friendly, menu-driven package that is very helpful for performing calculations related to common probability distributions and also for conducting small-scale simulations. Minitab is freely available in many PC labs on campus, and you can also download a free copy of Minitab for your PC from the Cal Poly portal. R is a more advanced, command-driven statistical software package that has a much more powerful programming language. We will use R to write programs for conducting more extensive simulation analyses. R is freely available for PC or Macintosh computers. Instructions for downloading both Minitab and R are available from our course web page.

Grading policies: Your course grade will be determined by the following components, with relative weights as indicated:

• Homework assignments (15%)
• Investigation assignments (15%)
• Exams (3; 20% for your lowest, 25% for the other two)

Homework assignments: Homework assignments will be given regularly, during most class sessions, and will typically be due at the beginning of one of the next two class meetings. Assignments will be posted on the course website. These assignments will typically involve several problems, intended to help you learn and apply the concepts and techniques presented that day. A sample of the problems on each assignment will be graded, and you may drop the lowest two of your homework scores from the calculation of your overall average.  You may work with one partner on these assignments, submitting one sheet with both names, provided that both of you contribute substantially to the work. You may also discuss problems with others, but each person (or pair) should write up solutions individually.

Investigation assignments: Investigation assignments typically ask you to analyze a more involved problem, often addressing multiple aspects of the problem. These investigations typically require use of software and often require writing computer code. These will be assigned occasionally, and you will have several days to complete these investigations. You may work with one other person on most investigations, submitting one report with both names, provided that both of you contribute substantially to the work. You may also discuss these problems with others, but you may not share computer code.

Word-processed reports are preferred to hand-written ones, and computer output should be integrated into the report. These investigation assignments will be posted on the course website. They are due when indicated on the assignment, which may not be during class time. You may not drop any scores on investigation assignments.

Exams: We will have three exams but no final exam. You may be excused from an exam only with a written medical excuse.  These exams will be open-book and open-notes.  You will be provided with preparation advice before each exam.  One thing to keep in mind is that interpretations and explanations will be as important as calculations.

I propose that we use the three hours saved by not having a final to give you two hours for each exam. The exams will be written to be completed in one hour, but you should find the extra time to be helpful because insights to solve probability problems sometimes take a while to emerge. You will be given a choice of taking the exam on Thursday from 11:10-1pm or from 4:10-6pm. (If neither of these times works for you, we’ll schedule an alternate time for that Thursday or the following day.) The exam dates will be:

• Thur April 22
• Thur May 13
• Thur June 3

Schedule: The following is a very tentative schedule, always subject to change:

Week	Dates	Topics	Sections of Text
1	March 29-30, April 1	Simulation, Basic probability rules	1.1, 1.2, 1.3
2	April 5 – 8	Counting rules, Conditional probability	1.4, 1.5, 1.6
3	April 12 – 15	Discrete random variables, Expectation, Variance	2.1, 2.3, 2.4
4	April 19 – 22	Binomial and other discrete distributions	3.1, 3.2, 3.3, 3.5
5	April 27 – 29	Markov chains	4.1, 4.2, 4.3, 4.5, 4.6, 4.7
6	May 3 – 6	Continuous random variables	5.1, 5.2, 5.3
7	May 10 – 13	Normal, exponential, and other distributions	6.1, 6.2, 6.3, 6.4
8	May 17 – 20	Sampling distributions, Central Limit Theorem	2.5, 2.6, 8.1, 8.2
9	May 24 – 27	Poisson process	7.2, 7.3
10	June 1 – 3	Reliability models; My favorite problem	10.1, 10.2

Some notes about this schedule:

• Holidays are Wed March 31 and Mon May 31.
• Class will be canceled for a furlough day on Mon April 26.
• We will complete the course at the end of week 10, because I will be away during the week of finals. (I have the position of Chief Reader for the AP Statistics program, and I will be supervising the grading of 120,000 exams that week!)

Advice: I offer the following Top Ten (plus one) list of very simple but often ignored pieces of study hints for your consideration:

1. Come to class.

• It’s very important to keep up with the material.

2. Participate in class.

• Or else why bother to come?

3. Print handouts in advance, take good notes.

• These should be very valuable to study from.

4. Ask questions.

• In class, and in office hours, and by email

5. Help each other.

• You can learn a lot by working together.

6. Start assignments early.

• Especially with investigation assignments.

7. Don’t get behind.

• The material will build on itself and go by fairly quickly.

8. Express yourself clearly.

• To avoid miscommunications about a tricky topic.

9. Have fun with the material.

• We’ll be studying lots of interesting and fun problems!

10. Take pride in your work.

• Do the best you can with everything that you do in the course.

11. Think!

• That’s what pursuing a Cal Poly education is all about!

A common theme emerges from this list: You are responsible for your own learning. As your instructor, I view my role as providing you with contexts and opportunities that facilitate the learning process. Please call on me to help you with this learning in whatever ways I can.