Instructor: Prof. Megan Owen
Phone: 718-960-7423
Office hours: 1:00-1:50pm and 3:40-4:30pm on Tuesdays and 3:40-4:30pm on Thursdays in Gillet 137E
Course time: Tuesday and Thursday 2:00-3:40pm, Gillet 305


A First Course in Probability, 8th or 9th edition, by Sheldon Ross. Publisher: Pearson.
This textbook has been used for this course for multiple years, so it should be possible to find a used copy.

Other free online textbooks on Probability:


All problem sets should be submitted on paper at the beginning of class on the due date. Late homework, including that submitted at the end of class, is not accepted.

Your grade for this course will be based on:

Homework 15%
In-class Quizzes 25%
Midterm 25%
Final Exam 35%
You must take and pass the final to pass the course. The lowest three quizzes and lowest three problem set grades will be dropped.


Academic Integrity Policy

You are encouraged to work together on the homework, but you should write up the solutions by yourself.


Date: Topics: Reading: Homework assigned: Due:
Tues 29 August
Review of syllabus and academic integrity code.
Introduction, permutations
Syllabus; Textbook sections 1.1-1.3; HW 1a: Chapter 1 #1,6,8,9 (from Problems section)
Thurs 31 August
Combinations Textbook section 1.4
Permutations vs Combinations
How to Understand Combinations Using Multiplication
Why do we multiply combinations?
Visualizing permutations and combinations
HW 1b: Chapter 1 #12,20,24 (from Problems section); #7 (from Theoretical Exercises)
Mon 4 September CUNY: No classes (Labor Day)
Tues 5 September
Multinomial coefficients and balls in urns Textbook sections 1.5-1.6 HW 2a: Chapter 1 #25,27,31 (from Problems section)
Thurs 7 September
Sample spaces and events Textbook sections 2.1, 2.2, 2.5
Visualizing set relationships
HW 2b: Chapter 2 #1,3,19 (from problems section) ; How many different ways are there to walk from Columbus Circle (at 59th St and 8th Ave.) to Grand Central Station (at 46nd St. and Park Ave.)? Assume you can only walk down Streets and Avenues. That is, assume you cannot walk along Broadway, through alleys, or cut through buildings. You may need to look at a map. Homework 1a and 1b due at beginning of class
Tues 12 September
Basic probability propositions Textbook sections 2.3, 2.4 HW 3a: Chapter 2 # 10, 13 (from problems section); #6, 13 (from Theoretical Exercises) Quiz on Sections 1.1-1.4 (HW 1a and 1b)
Thurs 14 September
Basic probability propositions cont'd Textbook sections 2.3, 2.4 HW 3b: Chapter 2 # 8, 15,19 (from problems section); #7 (from Theoretical Exercises) Homework 2a and 2b due at beginning of class
Tues 19 September CUNY: Classes follow a Thurs schedule
Tues 19 September
Conditional Probability Textbook section 3.2
Conditional probability explained visually
Another visualization of conditional probability
HW 3c: Chapter 3 # 1, 2, 4, 17 (from problems section) Quiz on Sections 1.5, 1.6, 2.1, 2.2, 2.5 (HW 2a and 2b)
20-22 September CUNY: No classes
Tues 26 September
Bayes's Formula and Independent events Textbook sections 3.3 and 3.4
Explanation and intuition of Bayes' Formula
HW 4a: Chapter 3 #19, 35, 59 (from problems section); 2 (from theoretical exercises) Quiz on Sections 1.5, 1.6, 2.1, 2.2, 2.5 (HW 2a and 2b)
29-30 September CUNY: No classes
Thurs 28 September
Independent Events Cont'd, Introduction to Random Variables Textbook sections 3.4, 4.1 HW 4b: Chapter 3 #33, 55, 62 (from problems section); 9 (from theoretical exercises) Homework 3a, 3b, and 3c due at beginning of class
Tues 3 October
Introduction to Random Variables Textbook section 4.1
Visualizing random variables
HW 5a: Chapter 4 #1, 5, 6, 7 (from problems section) Quiz on Sections 2.3, 2.4, and 3.2 (HW 3a, 3b, and 3c)
Thurs 5 October
Discrete Random Variables, Expectation of a Random Variable Textbook sections 4.2, 4.3, 4.4
Visualizing expectation
HW 5b: Chapter 4 #18, 19, 21, 25 (from problems section) Homework 4a and 4b due at beginning of class
9 October CUNY: No classes (Columbus Day)
Tues 10 October
Variance of random variables Textbook sections 4.4, 4.5
Visualizing variance
HW 6a: Chapter 4 #28, 35, 37 (from problems section); #8 (from theoretical exercises) Quiz on Sections 3.3 and 3.4 (HW 4a and 4b)
Thurs 12 October
Variance Continued, Bernoulli and Binomial Random Variables Introduced Textbook sections 4.5 and 4.6 HW 6b: Question 1: Chapter 4 #38 (from problems section)
Questions 2 and 3: Let X and Y be discrete random variables with the same possible values. Prove the following using the definition from class of expected value of a random variable or a function of a random variable:
(i) E[X + Y] = E[X] + E[Y]
(ii) E[aX^2] = aE[X^2]
Question 4: Let B be a Bernoulli random variable with p = 0.4. What is E[B] and Var(B)?
Homework 5a and 5b due at beginning of class
Tues 17 October
Binomial Random Variables continued Textbook sections 4.6
Visualization of binomial distribution
HW 7a: Chapter 4 #40, 41, 59 (submit on or by beginning of class on Thurs. Oct. 26 for extra credit) Note: #44 has been removed Quiz on Sections 4.1, 4.2, 4.3, 4.4 (HW 5a and 5b)
Thurs 19 October
Review for midterm Homework 6a and 6b due at beginning of class
Tues 24 October
Midterm Exam
Thurs 26 October
Binomial random variables continued, Poisson random variables Textbook sections 4.6, 4.7
Visualizing Poisson and other distributions
HW 7b:
Tues 31 October
Went over midterm HW 8a: Chapter 4 #48, 51, 55, 58 (from problems section) No quiz
Thurs 2 November
Poisson randome variables continued, introduction to continuous random variables Textbook sections 4.7, 5.1
HW 8b: Chapter 5 #1, 3, 4, 13 (from problems section) Note: The cumulative distribution function for continuous random variables is defined the same way as for discrete random variables:
F(a) = P{X ≤ a}
Homework 7a and 7b due at beginning of class
Tues 7 November
Expectation and Variance of Continuous Random Variables, Uniform Random Variable Textbook sections 5.1, 5.2, 5.3 HW 9a: Chapter 5 #2, 5, 6, 7 (from problems section) Quiz on Sections 4.6 (HW 7a and 7b)
Thurs 9 November
Uniform Random Variable, Normal Random Variable, Exponential Random Variable (if time) Textbook sections 5.3, 5.4, 5.5 (if time)
Normal distribution interactive graph
Normal distribution calculator
HW 9b: Chapter 5 #10, 14 (Prop. 2.1 is on page 191), 17, 37 (from problems section) Homework 8a and 8b due at beginning of class
10 November Last day to withdraw from class with a grade of W
Tues 14 November
Normal random variable continued, Jointly distributed random variables Textbook section 5.4 HW 10a: Chapter 5 #15, 16, 19, 21 (from problems section) Quiz on Sections 4.7, 5.1 (HW 8a and 8b)
Thurs 16 November
Jointly distributed random variables Textbook section 6.1 HW 10b: Chapter 6 # 1, 2a, 3a, 4 (from problems section) Homework 9a and 9b due at beginning of class
21 November Classes follow Friday schedule
23-25 November Thanksgiving Recess: College Closed
Tues 28 November
Jointly distributed random variables, independent random variables Textbook sections 6.1 and 6.2 HW 11a: Chapter 6 #6, 9, 23 (a, b, and c only; hint: to find E[X], find the marginal distribution of X), 40b (from problems section) Quiz on Sections 5.1, 5.2, 5.3 (HW 9a and 9b)
Thurs 30 November
Expectation of Sums of Random Variables, Chebyshev's Inequality and the Weak Law of Large Numbers Textbook section 7.2, 8.2
Visualization of Weak Law of Large Numbers (2nd figure)
HW 11b: Chapter 8 #1, 2 (from problems section); Let X and Y be independent random variables. Prove that E[XY] = E[X]E[Y] when a) X and Y are discrete random variables with joint pmf p(x,y), and b) X and Y are continuous random variables with joint pdf f(x,y). Note that this is not true when X and are Y are dependent. Homework 10a and 10b due at beginning of class
Tues 5 December
Central Limit Theorem (CLT) Textbook section 8.3
Visualization of CLT for Bernoulli r.v.
Visualization of CLT from statistics perspective
Another visualization of CLT
HW 12a: Chapter 8 #6, 13ab, 15, 16 (from problem section) (I recommend doing #6 last) Due: Tues. Dec. 12 Quiz on Sections 5.4 and 6.1 (HW 10a and 10b)
Thurs 7 December
Markov chains (not on final exam) and review for final exam Markov Chains explained visually Homework 11a and 11b due at beginning of class
Tues 12 December
Review for final exam Homework 12a due at beginning of class
Thurs 14 December Final exam 1:30pm - 3:30pm