*How dare we speak of the laws of chance?
Is not chance the antithesis of all law?*

--Bertrand Russell

**Final Update**: The grades have been posted. Have a great summer!

Winter 2010

**Instructor:**John Stembridge**Office:**4854 East Hall**Phone:**936-1790**email:**jrs AT umich DOT edu**Office Hours:**M 11-12:30, F 2:00-3:30

We will cover most of Chapters 1-7 and a little of Chapter 8. I will be expecting you to read the relevant sections of the book, and not merely use the text as the place where you look up homework problems. But keep in mind that some of the sections include applications that are too advanced for this course. I'll try to give you specific alerts from time to time.

**Prerequisite:**
Math 215 or 285 (multi-variable calculus).

The first month or so will be devoted to discrete probability, so it is easy
to get lulled into thinking that you can get by without all of the calculus
you studied years ago. If you don't remember how to deal with double
integrals or related concepts, you'll need to review.

**Homework is important!**

There will be approximately 10 posted problem sets (see below).
You can turn in assignments either in class or at my office
(4854 East Hall) by 4:00pm of the date due--usually a Monday.
The first homework set will be due on **Friday, January 15**.
Depending on the number of hours allotted to the grader(s) for these
sections, it is possible that not all assigned homework problems will
be graded.

** NO LATE HOMEWORK!**

**Exams:**

- First Hour Exam: Wed., Feb. 10
[Solutions]

Exam 1 results: Average 77.8; median 80; std. dev. 14.3. - Second Hour Exam: Wed., Mar. 24
[Solutions]

Exam 2 results: Average 79.8; median 82; std. dev. 13.4. - Final (Section 1): Tue., Apr. 27, 1:30-3:30
- Final (Section 2): Wed., Apr. 28, 10:30-12:30

Final results: Average 143.5/200; median 146; std. dev. 27.7.

**Grades:** will be weighted as follows

40%(Final) + 20%(Exam 1) + 20%(Exam 2) + 20%(Homework)

Your lowest homework set score will be dropped.

**Problem Set 1** / due Fri., 15 January.

- Describe your interests (academic or otherwise), or tell me something that makes you unique.
- Indicate your birthday (month/day) so I can use the data for an in-class experiment.
- Chapter 1 Problems (pp.16-17): 10, 20, 21, 27, 32.
- Chapter 1 Theoretical Exercises (pp.18-20): 11, 13, 18
(note:
**n=n**)._{1}+...+n_{r}

- Chapter 2 Problems (pp.50-54): 12, 15acd, 28, 37, 46.
- Optional (bonus points): 26.
- Chapter 2 Theoretical Exercises (pp.54-56): 6acghi, 12, 15.

- Chapter 2 Problems (pp.50-54): 53 [use Prop.4.4], 54 [what is a void?].
- Chapter 3 Problems (pp.102-110): 16, 23, 36, 40ab [use the Multiplication Rule (p.63) and/or (3.1)].
- Chapter 3 Theoretical Exercises (pp.110-113):
3 [assume
**k=2**for simplicity], 4.

- Chapter 3 Problems (pp.102-110): 55, 58, 60, 66 [Typo: use Figure 3.4], 74 [Hint: condition on the outcomes of the first rolls of A and B].
- Optional (bonus points): 80b [Warning: part (c) has a typo in
it --
**P**, not_{n}=1-P(E)**P(E)**.]. - Chapter 3 Theoretical Exercises (pp.110-113): 6, 10, 18.

- Chapter 4 Problems (pp.172-179): 4, 22, 29, 35, 42.
- Chapter 4 Theoretical Exercises (pp.179-183): 6, 13 [Hint: it's easier to maximize
**log P(X=k)**], 32.

- Chapter 4 Problems (pp.172-179): 17, 56 [use a Poisson approximation], 60, 65.
- Chapter 4 Theoretical Exercises (pp.179-183): 16.
- Optional (bonus points): 10 [p. 180].
- Chapter 5 Problems (pp.224-227): 4, 7.
- Chapter 5 Theoretical Exercises (pp.227-229): 5.

- Chapter 5 Problems (pp.224-227): 16, 22, 26, 31, 34.
- Chapter 5 Theoretical Exercises (pp.227-229): 13, 15, 31.

- Chapter 6 Problems (pp.287-291): 7, 9bce, 18, 20 [Hint: Example 2f], 26b, 29.
- Chapter 6 Theoretical Exercises (pp.291-293): 5a, 9 [Hint: What
is
**P(X_1 &ge a, ... , X_n &ge a)**and why is it relevant?].

- Chapter 6 Problems (pp.287-291): 38, 41.
- Problem A: Suppose
**(X,Y,Z)**have a multinomial distribution with parameters**n**and**p,q,r**, where**p+q+r=1**(see Example 1f on p.240). What is the conditional probability mass function for**X**given**Y**? That is, determine**P(X=i|Y=j)**. - Chapter 7 Problems (pp.373-379): 5, 8, 9, 21.
- Chapter 7 Theoretical Exercises (pp.380-384): 2.

- Chapter 7 Problems (pp.373-379): 36, 38, 42, 50, 57, 66.
- Chapter 7 Theoretical Exercises (pp.380-384): 17, 22.

- Chapter 7 Theoretical Exercises (pp.380-384): 45, 49.
- Chapter 8 Problems (pp.412-414): 5, 9.

The red curve above shows the probability of at least one match (two people sharing the same birthday) among N people, for N up to 70. The green curve shows the probability of more than one match.

The above graphic displays the probability mass functions for Binomial random variables with parameters

This page last modified Fri Apr 30 16:17:55 EDT 2010