CAS CS 237 - Probability in Computing - Fall 2025

Course Staff

Instructors Prof. Sofya Raskhodnikova
Prof. Tiago Januario
Teaching Fellows Anatoly Zavyalov
Erick Jimenez Berumen
Course Assistants Steve Choi
Letitia Caspersen
Daniel Matuzka
Vi Tjiong
Sarah Yuhan
Yoon Oh

Communication and Office hours

  • Piazza is the primary platform for all online discussions, questions, and answers.
    • Use private posts for sensitive or specific questions regarding your solutions or personal matters.
    • Do not send emails to the course staff.
  • We aim to maintain a positive and inclusive atmosphere on all course platforms.
  • If you need special accommodations or are facing unusual circumstances during the semester, please reach out to an instructor privately via Piazza as early as possible.
  • Your suggestions for improving the course are always encouraged and appreciated.

Prerequisites

We assume good working knowledge of elementary set theory and counting, elementary calculus (i.e., integration and differentiation), and programming in Python.


Syllabus

Introduction to basic probabilistic concepts and methods used in computer science. Develops an understanding of the crucial role played by randomness in computing, both as a powerful tool and as a challenge to confront and analyze. Emphasis on rigorous reasoning, analysis, and algorithmic thinking. This course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.


Course structure

  • Attendance in lectures and discussion is mandatory
  • The two lecture sections of the course will be treated as one class.
  • The content of the two lectures is identical, and assignments will be shared.

Textbooks

You can access both books for free or support the authors by purchasing the books.


Schedule

This schedule is subject, and likely, to change as we progress through the semester. Reading chapters are from the first textbook (LLM) or from the second textbook (P), referred to by the acronyms of the author names.

Date / Lec Agenda (Topics, Readings, Homework) Instr.
Lec 1
Tue, Sep 02
Course information
Tips to succeed
Random experiments
Sample spaces, events

[Slides with notes]
Read: P 1.1, P 1.2, OB 1B
TJ & SR
Lec 2
Thu, Sep 04
Probability function
Symmetry
Probability axioms

[Slides with notes]
Do: hw01 out
TJ
Lec 3
Tue, Sep 09
Probability rules
Computing probabilities

[Slides with notes]
SR
Lec 4
Thu, Sep 11
Tree diagrams
The Monty Hall problem

[Slides with notes]
Do: hw02 out
SR
Lec 5
Tue, Sep 16
Continuous Probability Spaces
Anomalies with Continuous Probability

[Slides with notes]
Read: P 1.3.5
TJ
Lec 6
Thu, Sep 18
Random variables
Sum of random variables
Definition and examples

[Slides with notes]
Do: hw03 out
TJ
Lec 7
Tue, Sep 23
Distribution Functions: PDF and CDF
[Slides with notes]
Watch: Video
TJ
Lec 8
Thu, Sep 25
Properties of PDFs and CDFs
Functions of random Variables

[Slides with notes]
Read: P 3.1.6, P 4.1.4
Do: hw04 out
Watch: Video
TJ
Lec 9
Tue, Sep 30
Conditional Probability
Pr(⋅ ∣ 𝐸) is a probability function

[Slides with notes]
Read: LLM 18, P 1.4.0
SR
Lec 10
Thu, Oct 02
Conditional Probability in Tree Diagrams
Product Rule
Independent Events

[Slides with notes]
Read: P 1.4.2, P 1.4.3
Do: hw05 out
SR
Lec 11
Tue, Oct 07
Law of Total Probability (generalization)
Bayes’ Rule - Part I

[Slides with notes]
Last Day to Drop Standard Courses (without a “W” grade)
SR
Lec 12
Thu, Oct 09
Bayes’ Rule - Part II
Independence for Multiple Events

[Slides with notes]
Read: LLM 18.9, P 1.4.1
Do: hw06 out
Watch: Video
SR
Tue, Oct 14
Substitute Monday Schedule of Classes
Check the Google Calendar for the updated office hour schedule
Lec 13
Thu, Oct 16
Pairwise and Mutual Independence
Independence for Random Variables

[Slides with notes]
Read: LLM 19.4, P 3.2.2
Do: Midterm practice problems out
SR
Lec 14
Tue, Oct 21
Read: LLM 19.5, P 6.1.2
Do: Midterm practice solutions out
SR
Thu, Oct 23
Midterm, in class, during lecture time Covers all topics up to Lecture 13
Do: hw07 out
Lec 15
Tue, Oct 28
Expected value of a random variable
[Slides with notes]
TJ
Lec 16
Thu, Oct 30
Infinite sums
Linearity of expectation
Read: LLM 20.3, P 3.2.4
Do: hw08 out
Watch: Video
TJ
Lec 17
Tue, Nov 04
Law of the unconscious statistician
Conditional expectation
Linearity of conditional expectation
TJ
Lec 18
Thu, Nov 06
Law of total expectation
Variance
Standard deviation
Variance properties
Read: LLM 19.5.4
Do: hw09 out
TJ
Lec 19
Tue, Nov 11
Discrete distributions: Bernoulli, Uniform, Binomial
Read: Wikipedia
SR
Lec 20
Thu, Nov 13
Discrete distributions: Geometric and its properties
Do: hw10 out
SR
Lec 21
Tue, Nov 18
Coupon collector's problem
Reservoir sampling
Negative Binomial
SR
Lec 22
Thu, Nov 20
Markov inequality
Chebyshev inequality
Do: hw11 out
TJ
Lec 23
Tue, Nov 25
Applications of Markov and Chebyshev's inequalities
Continuous Uniform Distribution
Read: P 4.2.2, P 11.1.2
TJ
Thu, Nov 27 Thanksgiving break
Lec 24
Tue, Dec 02
Normal Distribution
Read: P 11.1.2
TJ
Lec 25
Thu, Dec 04
Exponential Distribution Course evaluation
Read: CLRS 8.4
Do: hw12 out
Final Practice Problems out
SR
Lec 26
Tue, Dec 09
Poisson Distribution Course evaluation
Read: Wiki
Do: Final Practice Solutions out
SR
Thu, Dec 11 Study period begins
Final Exam
Dec 18
Final Exam Time: 3:00 PM – 5:00 PM
Location: HAR 105
Cumulative and covers all topics

We have prearranged the following date and time for students with final exam conflict:

  • Alternative Final exam date: December 16th
  • Time: 3:00 PM – 5:00 PM
  • Location: STHB19

Students with exam accommodations, regardless or their final exam conflict, will take the final exam at same day and initial time of the other students at CDS 801.


Participation and Attendance

  • Participation in lecture will be tracked with Top Hat questions.
  • Participation in discussion labs will be tracked with Top Hat location-based attendance.
  • You will get the full participation points if you answer at least 85% of the possible Top Hat questions.
  • You will get the full attendance points if you attend at least 85% of the discussion labs.
  • If you end up with x% points, where x < 85, you will get x/85 of the points.
  • Most of the material covered in lectures and labs can be found in our textbooks. Read them!
  • While our textbook will be very helpful, it is an imperfect substitute for in-class learning, which is the fastest (and easiest) way to learn the material.
  • In all cases, you are responsible for being up to date on the material.

Homework Policy

  • Submission: Weekly assignments are posted on Thursdays and due by Wednesdays at 9:00 PM ET via Gradescope. A 3-hour grace period is allowed; submissions after this period will not be accepted. Missing submissions will be treated as late.
  • Content: Provide step-by-step explanations for your answers. Submissions with only answers will receive minimal credit.
  • Formatting: Submit solutions as one single PDF file with high-quality images. Illegible submissions will receive a 0. We recommend using Dropbox for scanning.
  • Gradescope Page Selection: Correctly select the pages for each problem on Gradescope. Failure to do so will result in a 10% penalty. If you don't have a solution, note "No solution provided."
  • Late Policy: You may use up to 3 grace periods without penalty. After that, each late submission incurs a 1% penalty. The lowest homework grade will be dropped after penalties are applied.
  • Academic Integrity and Collaboration:
    • The Collaboration & Honesty Policy outlines the rules of collaboration and penalties for cheating. All students are required to read, sign, and submit this document to Gradescope.
    • Submitting identically worded answers, including pseudocode, is a serious offense and will be reported to the Dean's Office (BU Academic Conduct Code).
    • Using ChatGPT or similar AI for homework solutions violates the Collaboration & Honesty Policy.
    • You are not required to cite material from the course textbook, lectures, discussion notes, or information provided by course staff.
    • For any other external information, a proper citation is required. Failure to cite constitutes plagiarism.
    • Explicitly searching for problem answers online or from individuals not enrolled in the current semester's class is strictly forbidden.
    • If you receive help on Piazza or during office hours from instructors for specific problems, you must list them as collaborators.
  • Partial Submissions: Submitting partial work is acceptable if you cannot fully complete an assignment; avoid missing the deadline entirely.

Exams

  • Both exams will consist of problem-solving and short questions about the material.
  • Each exam duration and their locations are given in the course schedule.
  • The content of the final is cumulative.
  • No collaboration whatsoever is permitted on exams, any violation will be reported to the College.

Grading

The course grade will break down as follows:

  • 5% class participation
  • 5% lab attendance
  • 25% weekly homework assignments
  • 30% in-class midterm exam
  • 35% in-class final exam. Don't make any travel plans before the final date is released.
  • Incompletes for this class will be granted based on CAS Policy.
  • Note: Optional homework problems do not count towards your course grade (except when you are right on the border), but we will look at how you did on them if you ask for a recommendation letter.

Regrade Policy

  • Regrade requests can be submitted up to one week (7 days) after grades for a given assignment have been posted (except the final exam).
  • You must request a regrade via Gradescope, NOT through email.
  • When we regrade a problem, your score may go up or down.

Miscellaneous

LaTeX resources

Homework template files: tex, pdf, jpg.