Math 20

Discrete Probability

Instructor: Carl Pomerance (

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


Welcome to Math 20! (that's not a factorial sign...)

Week of 6/1: The x-hour will be met this week.
Regularly scheduled office hours on Tuesday, Wednesday, Thursday at 9 AM. As usual, you can schedule at another time.
Our final exam will be Satruday at 8 AM in Kemeny 007. It will be comprehensive over the entire term.
I should be able to inform you later this week what your letter grade would be if the final exam grade were dropped.
(I gave out these grades in the x-hour Tuesday.)

Here is a proof of Theorem 10.2 in the book that does not use Vandermonde determinants.

Here are solutions from our second test.

Here is the second midterm from when I taught Math 20 ten years ago. (This old exam has some Chapter 9 questions.)
Here are solutions of the old test.

Here is our first test, with solutions.

Here is the first test I gave in Math 20 when I taught it 10 years ago.


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.


Classroom, 108 Kemeny
Lectures: Monday-Wednesday-Friday 11:15am--12:20 pm (11 hour)
X-hour: Tuesday 12:00 pm--12:50 pm


Carl Pomerance -- 339 Kemeny / Tel. 6-2635
Office hours: Tuesday, Wednesday, Thursday 9:00 am--10:00 am and by appointment at other times.
Patty Neckowicz


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.

We will cover most of this book, but emphasizing discrete probability. Week one will cover chapters 1 and 2.


Homework 20%, two mid term exams each 25%, final exam 30%. 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.


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. Look under the "News" setting for the current assignment and under the "Past assignments" heading for prior assignments.

Past assignments

Homework due Monday, April 6: Section 1.2, problems 6, 8, 12, 14, 16, 26, 28.

Homework due Monday, April 13:
Section 3.1, problems 2, 6, 12, 22a (for this problem, find the probability that there are 56+j watches, and show this is maximal for j = 0).
Section 3.2, problems 2, 6, 8, 18.

Homework due April 20: Section 4.1, problems 2, 6, 12, 14, 16, 18, 22.
Also: Let n be a large integer. What is the approximate probability that a random permutation from {1,2,...,n} to {1,2,...,n} has at least 2 fixed points, given that it has at least 1 fixed point?

Homework due Monday, 4/27:
Section 5.1, numbers 6, 8, 10 14, 16, 18, 20, 24.
(On problem 10, assume the two censuses are independent events, and to get thinking right, you might look at problem 9.)

Homework due Monday, May 4: Section 6.1, numbers 2, 4, 6, 8, 14, 18, 22, 36. (In 22, the reference should be to Exercise 1.1.13.)

Homework due Monday 5/11:
Section 6.2, numbers 4, 8, 10, 12;
Section 7.1, number 2;
Section 8.1, numbers 6, 8, 10. (In number 10, assume k > 0.)

Homework due Monday, May 18:
Section 9.1, numbers 2, 4, 6, 8, 10, 12; Section 9.2, numbers 2, 6.

Homework due Wednesday, May 27:
Sec. 10.1, numbers 4, 6,
Sec. 10.2, number 2
Sec. 11.1, number 2.

Homework due Monday, June 1:
Section 12.1, numbers 2, 4, 6.


The two midterm exams will be held in the evenings of April 21 and May 12 from 7:00pm to 9:00pm in Carson L01. Hopefully these exams will 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 6 from 8:00 am to 11:00 am in Kemeny 007.

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 their 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.