Mathematical Theory of Probability (477)

In a nutshell


Contact: Office: Hill 432, Email: jwolf137 at math dot rutgers dot edu
Lectures: Monday and Thursday, 12:00-1:20pm, ARC-205, BUS.
Credits: 3 credits (for at most one out of 01:640:477, 01:198:206, 01:960:381, or 14:330:349).
Prerequisites: Math 251 (third semester multivariable calculus)
Links: department course information
Textbook: Sheldon Ross, A first course in probability, 7th Edition, Prentice-Hall.
Office hours: Monday 10:30-11:30am at Hill 432 and by appointment.
Homework assignments: announced on Sakai, due at the beginning of each lecture.



If there were one mathematics course that should be made compulsory for ANYONE's education, this would be it. Apart from representing beautiful and elegant mathematics, probability is absolutely essential for being able to make sense of the world around you. In fields as diverse as politics, economics or the life sciences, probabilistic models are ubiquitous and extremely powerful. This course hopes to lay the foundations and provides you with the basic tools needed to make informed choices.


The following list of topics covered is tentative so please check back frequently. General course announcements will be made via the course's Sakai page.

Thursday Jan 221.1 - 1.3 introduction
MondayJan 261.3 - 1.5 basic counting: permutations and combinations
Thursday Jan 292.1 - 2.3 sample spaces, events and the axioms of probability
Monday Feb 21.6, 2.4 the inclusion-exclusion principle and some number theory
Thursday Feb 52.5 uniform distributions and examples
Monday Feb 93.1 - 3.3 conditional probability and Bayes' formula
Thursday Feb 123.4 independent events
Monday Feb 163.4 - 3.5 repeated independent trials
Thursday Feb 194.1 - 4.2 random variables and distribution functions
Monday Feb 234.3 - 4.5 expectation and variance of discrete random variables
Thursday Feb 264.6 - 4.7 the bernoulli, binomial and poisson distributions
Monday Mar 2 ***SEVERE SNOW***
Thursday Mar 5 ***MIDTERM EXAM NO. 1***
Monday Mar 94.8.1 - 4.8.3 the negative binomial, geometric and hypergeometric distributions
Thursday Mar 125.1 - 5.2 expectation and variance of continuous random variables
Monday Mar 16 ***SPRING BREAK***
Thursday Mar 19 ***SPRING BREAK***
Monday Mar 235.3 - 5.5 the uniform, normal and exponential distributions
Thursday Mar 265.6 - 5.7 the gamma and beta distributions, distribution of a function of a random variable
Monday Mar 306.1 jointly distributed random variables
Thursday Apr 26.2 independent random variables
Monday Apr 66.3 sums of independent random variables
Thursday Apr 9 review
Monday Apr 13 ***MIDTERM EXAM NO. 2***
Thursday Apr 166.4-6.5 conditional distributions
Monday Apr 207.1-7.4 properties of expectation, covariance and correlation
Thursday Apr 237.5 conditional expectation
Monday Apr 277.7 moment generating functions
Thursday Apr 308.1 - 8.2 markov and chebychev inequalities, the weak law of large numbers
Monday May 48.3 the central limit theorem
Wednesday May 13 ***FINAL EXAM***

Homework assignments

Homework problems and grades will be announced on Sakai as we go along. You should expect to spend about 2 1/2 hours per lecture working on homework assignments and consolidating your knowledge of the material by re-reading your notes and following up with the textbook. Homework problems will be collected at the start of each lecture.
Note that late homework will NOT be accepted. Gaps between lectures are sufficiently long as to make this a legitimate rule.


There will be an average of one quiz per week of no more than 10 minutes' length. You should be able to answer the questions without too much trouble if you pay attention in class and keep up with the homework.
There will generally be NO opportunity to make up quizzes, so class attendance is crucial.


There will be NO calculators in this class. I am hoping to teach you to use your brain instead.

Grading information

A total of 500 points is available for this class, which will be allocated as follows.

Final exam:200
Midterm I:100
Midterm II:100
Homework:  50
Quizzes:  50

A note on the use of Sakai

While we try our best to accurately transfer your grades to the Sakai gradebook, mistakes do occur. It is your responsibility to check your grades and notify us of any errors no more than a week after they were first announced on Sakai.

Rules for the Midterm Exams

The exam will take place in ARC-205, where we hold lectures. We will start promptly at 12pm.

1) No notes or textbook material may be used during the exam.

2) You may bring a calculator, although I would strongly discourage you from actually using it.

3) NO other electronic devices may be used during the exam. This includes cellphones and music players of any kind. Cellphones MUST be turned off (not just silenced). If you do not have a watch and usually rely on your phone to tell the time, borrow one for the exam.

4) NO scrap paper of any type may be brought into the exam room. Paper for your rough calculations will be provided if necessary.

5) Points available per question as well as the total will be announced on the exam booklet.

6) It is YOUR responsibility to ensure that you have written your name at the top of every sheet that you turn in.

7) You will NOT be allowed access to the exam room more than 5 minutes ahead of time. PLEASE BE ON TIME. Once you have entered the room, do NOT turn over the cover page of your exam booklet until you are told to do so.

8) Only ONE person will be allowed to go to the bathroom at any one time. If you finish early please leave the room quietly. If you finish within the last 15 minutes we ask you to kindly stay until the end of the exam period to avoid total mayhem.

9) Actual and ATTEMPTED cheating is treated very seriously at Rutgers. Our academic integrity policy calls cheating in a midterm of final exam a "Level 3 violation", which could result in an F and a suspension of one or more semesters.You will have a different version of the exam from your neighbours.

10) A rescheduling of the exam is only possible in the case of serious illness, a major emergency, or a non-negotiable outside commitment. If the reason for your absence from the exam is known in advance you MUST ask for permission from the lecturer before the exam and provide appropriate evidence of your situation.
In all other cases, you must notify the lecturer by email at or through the Mathematics Department Undergraduate Office at 732 445-2390 as soon as possible. If you notify the lecturer LESS THAN ONE WEEK before the exam, then documentation verifying the reason for your absence (such as a doctor's note) needs to be presented to the appropriate Dean's office. A written note from the Dean requesting a make-up exam will have to be presented to the lecturer.

Tips for the exam itself

1) Write neatly and in the space provided. This will also help you come back to a question that you got stuck on.
2) Show your work - you may get partial credit even if you don't complete a question.
3) State any rules and formulae that you use clearly before applying them. Again, you may receive partial credit for doing so.
4) Use "=" and other symbols appropriately.
5) Label any diagrams you draw clearly.
6) Actually draw those diagrams!
7) Define your random variables and events. State clearly what distribution you are using and what its parameters are. 8) Think about your exam strategy in advance and work to your strengths.
9) Do not panic if you get stuck or if you get behind in time.

Materials to help you review

1) Your textbook is very good at providing a multitude of interesting examples. However, it's quite difficult to uncover precisely what we have been doing all this time (one of those "can't-see-the-forest-for-the-trees" situations). If you just want a VERY concise summary of the actual concepts we have covered, with few distracting examples, I recommend Prof. Kennedy's Probability IA lecture notes from Cambridge. We've now covered almost everything up to and including Section 4.3, with the exception of conditional distributions which we'll do formally after the exam.

2) If you want to see what past exams for 477 have looked like, have a look at Prof. Speer's page. Scroll all the way to the bottom to find his midterm exams from the fall semester with solutions. There are also review problems available from the same spot. Moreover, he maintains a separate page where he collects solutions to selected exercises from the book which might be useful to you.

3) Work through the examples we did in class for each topic. Do NOT look at the solutions without trying very hard to solve them yourself first. What we did in class is very representative of what is in my head, and the same holds for the exam.

3) Do as many practice questions from the book as you can, preferably those that have solutions so you can check your answers. Or try those covered my Prof. Speer's page, see 2) above.

4) We will have a review in class on Thursday. Please come prepared.

The Probability Challenge

In addition to homework and quizzes, I will be setting one Probability Challenge question per week, as per student request now on Monday due before class the following Monday. Answers should be submitted by electronic mail to jwolf137 at rutgers dot math dot edu.
These have more of a puzzle or brainteaser flavour and are mainly intended for your enjoyment. They do NOT count towards your course grade. However, if you are on the borderline between two grades at the end of the semester and have been doing consistently well on these, I am more likely to round up than down. Also, there will be a prize (to be determined, suggestions welcome) for the winner of this competition.

Code number  1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total
115006753 x x x x x 5
051002228 x x 2
066005495 x x x x 4
112000384 x 1
023009984 x 1
114002521 x x x x x 5
112005317 x x x x x 5
092005690 x x x x 4
114005272 x x x x x x x x x x x x x x 14
100007571 x x x x x 5
125000779 x x x 3
115005583 x 4
072000874 x x x x x x x x x x x x x 13
017006640 x x x x x 5
115007174 x x x x x x x x x x x x x x 14
123009554 x x x x x x x x x x x x 12
115001854 x x 2
113007262 x x x x x x x x x x x x x x 14
520131444 x x 2

Each "x" is worth 1 point. Each "♣" is worth 3 points. I'll try and update this page after every round. Please let me know by email if you think there is a mistake in this table.

This page was last updated 20th January 2009.