Math 20

Discrete Probability

Instructor: Carl Pomerance (

Abstract | Classes | Staff | Textbook | Grading | Homework | Past assignments | Exams | Honor Code


Our final exam will be 8 - 11 AM on Friday, June 3 in our classroom.
It will be cumulative over the course, with roughly a uniform distribution of material. The length will be longer than the midterms, but not twice as long.
A good way to study is to go over old homework and test problems, including looking at the handouts for problems that confuse you. In addition, try looking at odd-numbered problems in the book that were not assigned. In general, it is good to know why things are true, not just that they are true. This helps for remembering both short and long term. Finally, studying in a group is fine, but you should also do some on your own. It may be easy to fool yourself that you understand something when you're with others who may understand it better than you do.
Good luck!


Basic concepts of probability are introduced in terms of finite probability spaces. The basic theory is often introduced in terms of simple models such as coin tossing, random walks, and casino games.


Reed Hall, Room 106
Lectures: Monday-Wednesday-Friday 11:15am--12:20am (11 hour)
X-hour: Tuesday 12:00am--12:50am


Carl Pomerance -- 102 Choate House (corner of Choate and North Main)/ Tel. 6-2635
Office hours: Tuesday, Wednesday, Thursday 9:00am--10:00am and by arrangement at other times.
Fan Zhong


Introduction to Probability, second revised edition, by Charles M. Grinstead and J. Laurie Snell

This book is available from Wheelock Books
This book is also available for free on the web: .
Answers to odd-numbered problems are also found at that website.


Homework 20%, two mid term exams each 20%, final exam 40%. As much as possible, grades will be based on demonstrated knowledge. However relative performance may be used as a criterion for increasing grades, and grade borderlines will be chosen to place a relatively small number of students on borderlines. At the end of the term, the lowest of your 4 grades (hw, midterms, final) will be dropped, except if your final exam is your lowest grade, in which case the weight of the final exam will be halved. (So, if one of the midterms or hw is dropped, then the remaining 3 grades have weights 25, 25, 50; while if the final is the lowest, the four grades have equal weight 25, 25, 25, 25.)


Homework is due at the start of the class period on the due date. Late homework is generally not accepted unless there is a prior arrangement.

Past assignments

Homework due FRIDAY, May 27:
Section 10.2, number 2.
Section 11.1, numbers 2, 6 (interpret w^(n) as u^(n) as in Theorem 11.2)
Section 11.2, numbers 4, 6, 8.

Homework due Monday, May 23:
Section 10.1, numbers 2, 8.

Homework due Monday, May 16:
Section 9.1, numbers 2, 4, 6, 8, 10, 14.

Homework due Monday, May 9:
Section 6.2, any 10 problems from numbers 2, 4, 8, 10, 12, 14, 20, 22, 24, 28, 30.
(Do all 11 for extra credit.) Some students tell me that there is no number 30 in their texts for section 6.2. I believe in this case you should do number 29 instead: Compute the variance for the Poisson random variable with parameter &lambda and show that it is &lambda .
Section 7.1, number 2. (If you did number 4, as posted earlier, hand it in separately for extra credit.)
Section 8.1, numbers 8, 14. (For number 14, use the same method of proof as we did for Chebyshev's inequality.)

Homework due Monday, May 2:
Section 6.1, numbers 2, 4, 6, 8, 12, 14, 18, 30 a,c,d, 40.

Homework due Monday, April 25:
Section 5.1, numbers 6-8, 13, 14, 16-20, 24, 25, 28, 31, 34, 38.

Homework due Monday, April 18:
Section 4.1, numbers 2, 4, 6, 8, 12, 14, 16, 22, 26, 34, 36, 38.
Extra problem: I have a secret integer that I've chosen randomly between one and one million (inclusive). One million people independently try to guess my number. What is the probability that at least one person will guess right? What is the conditional probability that at least two people guess right, given that at least one person guesses right?

Homework due Monday, April 11:
Sec. 3.1, numbers 4, 6, 12.
Sec. 3.2, numbers 2, 6, 8, 10, 12, 20, 27, 32, 34a.

Homework due Monday, April 4:
Sec. 1.1, #14, #16. (By "simulate", the book means to try out some cases, perhaps by a computer program, perhaps just by thinking hard! These questions are not cookie-cutter, and are designed to have you think about some interesting issues. We'll learn precise tools to deal with this type of question later in the course.)
Sec. 1.2, #2-4, #6, #10.
Sec. 1.2, #12, #14, #16, #18b, #20, #28, #31.


The two midterm exams will be held in the evenings of April 19 and May 17 from 7:00pm to 9:00pm. These exams will be constructed so as to be doable in 60 minutes; the extra hour is to help you relax and not feel so rushed.

The final exam will be held on June 3 from 8:00am to 11:00am.

Honor Code

Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. Students are discouraged from using solutions to problems that may be posted on the web for previous offerings of the course, and as just stated, must reference them if they use them. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of anotehr person's code or solutions, in whole or in part, is a violation of the Honor Code.

The honor principle applies to exams as follows: Students may not give or receive assistance of any kind on an exam from any person except the professor or someone explicitly designated by the professor to answer questions about the exam. Students may not use a computer during an exam, but they may use a calculator to help with simple arithmetic.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. It is always easier to ask beforehand.


I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

The Student Disabilities Coordinator, Nancy Pompian, can be reached at 6-2014 if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.

This page was inspired by the web site for Math 19 in Fall 04, written by Alin Popescu