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Umich Math 425 Homework 5 Solutions

Math 425: Introduction to Probability

Fall 2010, Section 3

Course meets: Tuesday and Thursday 1:10-2:30 in 1084 East Hall.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297, fomin@umich.edu

Office hours: Tuesday 3:00-4:00 and Thursday 4:10-5:30 in 2858 East Hall. No office hours on 9/23, 10/14, 11/4, 12/9.

Grader: Yingying Tan, yings@umich.edu

Course homepage: http://www.math.lsa.umich.edu/~fomin/425f10.html

Text (required): Sheldon Ross, A First Course in Probability, 8th edition, Prentice-Hall, 2010.

Where can I buy this textbook? (also check out publisher's online store and international editions)

Prerequisites: Math 215 or 285 (Multi-variable calculus).

Course synopsis:
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. The material corresponds to most of Chapters 1-7 and part of 8 of Ross.

Grade will be based on two midterm exams (held during regular class time), 25% each; 20% homework; 30% final exam (comprehensive).
Your lowest homework set score will be dropped.

This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out. Every student with the total score of 90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).


There will be 10 problem sets. No late homework will be accepted. In each homework assignment, 5 problems will be graded. All answers should be justified by a sound argument. An answer lacking justification will receive no credit. Collaboration on the homework is fine, but each person is responsible for writing up her/his own solutions. Due dates shown below are tentative; actual due dates will be announced in class.

HW#1, due 9/14: Chapter 1, problems 3, 8(c), 12(a), 15, 22, 24, 26, 28, 30.
HW#2, due 9/21: Chapter 2, 12(a), 17, 21(b), 29(a), 32, 42, 43(b), 45, 47.
HW#3, due 9/28: Chapter 3, 4, 13, 16, 18(a), 26, 36, 47, 53.
HW#4, due 10/05: Chapter 3, 39, 48, 56, 62, 63(c), 66, 70, 71.
HW#5, due 10/21: Chapter 4, 4, 14, 22(b), 32, 38, 41, 43, 48.
HW#6, due 10/28: Chapter 4, 54, 64(b), 65(b); Chapter 5, 1, 5, 7, 11, 13(b).
HW#7, due 11/4: Chapter 5, 18, 19, 21, 24, 27, 32, 33.
HW#8, due 11/23: Chapter 6, 6, 8, 14, 27, 29(a), 30(a), 31(a).
HW#9, due 12/2: Chapter 6, 41(a), 58; Chapter 7, 9(a), 12, 19(a), 34(a).
HW#10, due 12/9: Chapter 7, 34(b), 39, 42 (1st part), 51; Chapter 8, 7, 13(a), 15.


Exams are closed book, closed notebook. You will be allowed to bring a 3-by-5 index card to the 1st midterm, two such cards to the 2nd midterm, and three cards to the final. One problem on each midterm exam will be taken directly from homework (perhaps with altered numerical values).

Past exams can be found on Math 425 pages by Dan Burns and Hugh Montgomery.

The midterm exams are held in class. No makeups will be given.

Exam #1 covers Chapters 1-3.

Practice problems for Exam #1 and answers to them. Another practice problem

Exam #2 covers Chapters 4-5, with the exception of Sections 4.8.3-4.8.4, 5.5.1, 5.6.

Practice problems for Exam #2: Self-Test Problems and Exercises in Chapter 4, 3, 13, 15; and in Chapter 5, 11, 12

Final exam covers the topics covered by the midterm exams (see above), plus Sections 6.1-6.5, 6.7, 7.1-7.2 (except 7.2.1-7.2.2), 7.4-7.5 (except 7.5.2-7.5.4), 8.1-8.3, and 9.1.

Practice problems for the final exam and answers to them.

Exam times

All exams will be held in the same room where the class meets.
Midterm exam dates shown below are tentative; actual dates will be announced in class.
First midterm: October 7.
Second midterm: November 9.
Final exam: December 15, 4-6 PM, in 1084 East Hall.
(Time of the final exam is determined by the Office of the Registrar.)

Time & Location


Office hours

Mon, Wed, Fri: 10-11, 1-2
(and by appointment)


[ Announcements ] [ Course Description ] [ Schedule ]



Course Description


Three term calculus sequence. The course will make essential use of the material of Math 116 and 215.


A first course in probability by Sheldon Ross (Seventh edition).


This course introduces students to useful and interesting ideas of the mathematical theory of probability. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. We will cover most of chapters 1-8 from the textbook.


There will be 2 midterms (in class), and a final. We will have 11 weekly homework assignments. Grades will be determined by the following weighting.

Midterm 1 20%
Midterm 2 25%
Final 30%



1Jan 7Basic Principles of Counting 1.1-3Notes
2Jan 9Binomial Coefficients 1.4-5NotesHomework 1 (due Jan 21)
3Jan 12Review of Chapter 1 Notes
4Jan 14Probability Models 2.1-2Notes
5Jan 16Axioms of Probabilities 2.3-4Notes
6Jan 21Sample spaces with equiprobable outcomes 2.5Notes

Birthday applet
Homework 2 (due Jan 28)
7Jan 23Inclusion/Exclusion Principle
Review of Chapter 2
8Jan 26 Conditional Probability 3.1-2

Monty Hall Paradox

9Jan 28Bayes Theorem3.3NotesHomework 3 (due Feb 4)
10Jan 30Independence3.4Notes
11Feb 2 Conditional probability function 3.5 Notes
12Feb 4Review of Chapter 3 Notes
13Feb 6Discrete Random Variables 4.1-2 Notes

Benford's Law
14Feb 9Expectation 4.3-4Notes

Sic Bo
15Feb 11Midterm 1 Chapters 1 - 3 Midterm 1 guideHomework 4 (due Feb 18)
16Feb 13Variance 4.4-5Notes
17Feb 16Bernoulli distributions 4.6 Binomial Calculator
18Feb 18Poisson distributions 4.7 Poisson Calculator
Homework 5 (due March 4)
19Feb 20Other distributions 4.8Notes
Spring Break, No class on week of Feb 23 - 27
20March 2Continuous random variables 5.1-2, 5.7 Notes
21March 4Uniform distributions 5.3NotesHomework 6 (due March 11)
22March 6Normal distributions 5.4Notes
23March 9Normal distribution 5.4Notes
24March 11Exponential distribution5.5NotesHomework 7 (due March 18)
25March 13Other continuous distributions
26March 16Joint distributions6.1
28March 18Independent R.V.s6.2Notes
28March 20Sums of R.V.s 6.3Notes
29March 23Sums of R.V.s6.3
30March 25Midterm 2Chapters 4-5Midterm 2 guideHomework 8 (due April 1)
31March 27Conditional distributions 6.4-5Notes
32March 30Joint probability distributions 6.7Notes
33April 1Review Chapter 6 NotesHomework 9 (due April 8)
34April 3Expectation 7.1-2Notes
35April 6Expectation 7.2Notes
36April 8Covariance7.4NotesHomework 10 (due April 15)
37April 10 Conditional expectation7.5-6 Notes
38April 13 Chebyshev's inequality and the Weak Law 8.1-2 Notes
39April 15Central Limit Theorem8.3NotesHomework 11 (due April 27)
40April 17 Strong Law of Large Numbers 8.4Notes
41April 20 Chapter 8 Review
April 22Exam Review
(Not a required class day)
April 27Final Exam
Monday, April 27

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