CS109: Probability for Computer Scientists

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Chris Gregg

Summer 2026

We are in a paradigm-changing world…

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 is a stepping stone to understanding this new world

Probability is the core concept in machine learning and artificial intelligence algorithms!

This class is an applied mathematics course on probability

You will be seeing and writing code (in Python)

We will build on many concepts that lead to machine learning

Today:

We will discuss course logistics

We will start to talk about sample spaces, event spaces, and the meaning of a probability.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Who is Chris Gregg

• Education:
• BS Electrical Engineering, Johns Hopkins
• MS Education, Harvard
• Ph.D. Computer Engineering, University of Virginia
• U.S. Naval Officer
• Seven years Active Duty
• 15+ years Reserves
• Cryptologist / Information Warfare Officer
• Lived in San Diego (2x WestPac deployments), Australia, Djibouti
• Teaching
• High School Phyiscs
• Brookline High School, Brookline, MA
• Pacific Collegiate School, Santa Cruz, CA
• Computer Science
• Tufts University
• Stanford University

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Who is Chris Gregg

• I love teaching, and my current research focuses on computer science education
• I have been teaching for 25 years, and cannot think of a better job
• I love tinkering, and electronics. Check out some of my typewriter tinkering (1) and (2)
• At Stanford, I have taught CS106A, CS106B, CS107, CS107E, CS181W, CS208E, CS298, and now (first time!), CS109
• If you are interested in teaching, or in CS education, please ask me about it!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Chris and Probability

• When I took probability in college, it was a very different course. When I took probability in graduate school, it was a very different course…
• Today’s CS109 is geared towards teaching you the math you need to succeed in learning machine learning and AI concepts.
• See all those names below? This course has a long history, with many instructors over the years.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Why is CS109 important?

We are seeing a huge surge in statistics, predictions, and probabilistic models shared through global news, governing bodies, and social media.

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The technological and social innovation we develop during this time will strongly influence how we solve interesting problems impacting the lives of countless people across the globe.

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National Weather

Service Alerts

https://www.weather.gov/

World Politics

https://abcnews.go.com/538

https://www.nytimes.com/

https://www.economist.com/

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Logistics

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Staff

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TA: Justin Choo (oliver88@stanford.edu)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Mechanics (light version: see the syllabus for details)

• Main website: https://cs109.stanford.edu
• Ed Forum: https://edstem.org/us/courses/99736/discussion
• Prerequisites
• CS106B (important, coreq ok):
• Recursion / Hash Tables / Binary Trees
• Programming

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• CS103 (not necessary):
• Proof techniques (induction) / Set Theory / Math maturity

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• Math 21 or equivalent (important, coreq ok)
• Differentiation / Integration /Basic facility with linear algebra (vectors)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Components

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Lecture: MTuWTh 10:30am-11:45am, CoDa B80

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Components

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Written portion:

for typesetting, Tutorial on CS109 website

Coding portion in Python:

Review Session TBD

Late Policy:

All students given 2-hour grace period (but 2-hours and 1 second is actually late).

On the pset app, you can grant yourself up to two late days per assignment, up to five total. After that, you must contact the head CA

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Components

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We will have more information about both exams as they near.

If you have an OAE accommodation, please fill out the following form and upload your letter: https://forms.gle/49uNkj5c2EG8TjVi9

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Components

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We will not have regular sections for in-person students. We will have a section for CGOE students.

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Class time will be spent on a short recap of the lecture slides, and then we will spend the bulk of the time working on practice problems covering the material.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 Contest

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• Announced mid-quarter, genuinely optional
• Boost final course grades after letter grade buckets have been determined
• Your baseline is CS109, and the sky is the limit.

• Some recent winners:
• The Probability of Curing Cancer: Will My Clinical Trial Succeed?
• Modeling Indexical Fields as Bayesian Networks
• StatTuring: Distinguishing between LLM and Human text
• Parka: A Mobile App for Early Parkinson’s Disease Detection

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

If you have questions:

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Q&A forum

All announcements

“Working” office hours

start on Wednesday

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Course Reader: https://probabilitycoders.stanford.edu/sum26

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

How this class will work:

• Because the summer quarter has a relatively low enrollment, we are going to be running the class in a modified “flipped” format:
• Prior to each lecture, lecture slides will be provided, which students are expected to go through.
• We will also provide a series of “LLM Prompt” instructions, which will allow students to dig into the material using an LLM. Students should follow those prompts, and then explore the material in as much detail as they would like. This will allow you to ask targeted questions to enhance your own learning. Have fun with it!
• During class time, we will spend about 20 minutes going over the material in a lecture-like format, and then we will spend the rest of the time working on a combination of practice problems, old midterm and final exam problems, and homework problems.
• For most of the classes, both Chris and Justin will be available to help with the problems. We will also spend time reviewing the solution to many of the problems.
• We will do our best to highlight when we are solving problems on the board so CGOE students can jump to those sections of the video. We can also go over those problems during the CGOE section.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The Story of Modern AI

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Modern AI

or, How we learned to combine

probability and programming

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Brief History

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Early Optimism in the 1950s

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Early Optimism in the 1950s

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“Machines will be capable, within twenty years, of doing any work a man can do.”

–Herbert Simon, 1952

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Underwhelming Results 1950s to 1980s

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The world is too complex

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The (Second!) AI Winter Started to Thaw in the late 1990s

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1997: Deep Blue

2005: Stanley

2011: Watson

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Progression was Incredible!

Speech Recognition is Almost Perfect Today!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The last remaining board game

Humans conquered by AlphaGo in 2015

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Translation is AI-driven (and amazing)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Self-Driving Cars (right outside in Palo Alto!)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

In 2022, everything changed, again:

AI that seems to understand language

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

We Now Live in a World of AI

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

There are problems that seemed virtually impossible for a computer a short time ago

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The Chihuahua or Muffin Challenge

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The Chihuahua or Muffin Challenge

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The Chihuahua or Muffin Challenge

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

This is what a human sees: This is what a computer sees:

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Why is it easy for humans?

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About 30% of your cortex is used from vision

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3% is used to process hearing

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Let’s start a bit easier…digit recognition:

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Classification: Make a Harry Potter Sorting Hat

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Classification: Make a Harry Potter Sorting Hat

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That is an image of a one

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Classification: Make a Harry Potter Sorting Hat

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That is an image of a zero

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Classification: Make a Harry Potter Sorting Hat

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That is an image of a zero

* Sometimes it will be wrong!

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Classification is Hard to Program

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

In this class, we will learn the math that underpins classification

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

So all of probability is to inform machine learning?

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

So all of probability is to inform machine learning?

No.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Probability is more than just machine learning!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Algorithms and Probability

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E.g., Raytracing E.g., HashMaps

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Understanding the world and building tools

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E.g., CT Scanner: uses probability to determine if body material is bone, muscle, or fat.

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Philosophy and Ethics

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Most Desired Skill by PhD Students

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Most CS PhD students list their highest desiderata upon

graduation as:

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“Better understanding of probability”

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Learning Real Skills in CS109

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Learning Real Skills in CS109

Learning Real Skills in CS109

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Recent exam question:

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Patient sees a series of letters of different font size, and for each, answers correct or incorrect

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Learning Real Skills in CS109

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Recent exam question (part 2):

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A patient has answered 20 “letter sizes” and got a few

correct. What is your belief in how well they can see?

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Learning Real Skills in CS109

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Now state of the art for eye exam theory!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Learning Real Skills in CS109

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 Provides a Foundation for Your Future

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 Provides a Foundation for Your Future

But, it is not always intuitive

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Unintuitive Probability

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Unintuitive Probability

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• 0.8% of people have zika
• Test has 90% positive rate for people with zika
• Test has 7% positive rate for people without zika

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Unintuitive Probability

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• 0.8% of people have zika
• Test has 90% positive rate for people with zika
• Test has 7% positive rate for people without zika

The right answer is 9%

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 View of Probability:

Teach you how to write programs to numerically solve probability problems

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 View of Probability:

Teach you the theory you need to do the math

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

CS109 Major Topics

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Let’s Dive in to Some Probability!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Events and the Sample Space

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• Event, E, is some subset of S {E ⊆ S}

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• Coin flip is heads: E = {Head}
• ≥ 1 head on 2 coin flips: E = {[H, H], [H, T], [T, H]}
• Roll of die is 3 or less: E = {1, 2, 3}
• # emails in a day ≤ 20: E = {x | x ∈ ℤ, 0 ≤ x ≤ 20}
• Wasted day {≥ 5 YT hrs.}: E = {x | x ∈ ℝ, 5 ≤ x ≤ 24}

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Note: When Ross uses: ⊂, he really means: ⊆

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Events and the Sample Space

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

A probability is a number between 0 and 1

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Our belief that an event E might occur

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P(E)

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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The event we care about

How many times does it occur?

Out of (close to) infinite trials

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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n is the number of trials

The “event” E is that you hit the target

Hit: 0

Thrown: 0

P(E)≈

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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n is the number of trials

The “event” E is that you hit the target

Hit: 0

Thrown: 1

P(E)≈ 0.00

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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n is the number of trials

The “event” E is that you hit the target

Hit: 1

Thrown: 2

P(E)≈ 0.50

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability?

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n is the number of trials

The “event” E is that you hit the target

Hit: 11

Thrown: 24

P(E)≈ 0.46

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Let’s look at the Course Reader:

https://probabilitycoders.stanford.edu/spr26/probability

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability (in a Dataset)?

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Let E be the event that it is Sunny

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

What is a probability (in a Dataset)?

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Let E be the event that it is Sunny

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Equally Likely Outcomes

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Some sample spaces have equally likely outcomes.

• Coin flip: S = {Head, Tails}
• Flipping two coins: S = {[H, H], [H, T], [T, H], [T, T]}
• Roll of 6-sided die: S = {1, 2, 3, 4, 5, 6}

If we have equally likely outcomes, then P(Each outcome)

Therefore

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Not Everything is Equally Likely

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Not Everything is Equally Likely

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• Play lottery.
• What is P(Win)?

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Not Everything is Equally Likely

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• Play lottery.
• What is P(Win)?

• S = {Lose, Win}
• E = {Win}
• P(Win) = |E|/|S| = 1/2 = 50%

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Solve probabilities using math!

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

The Axioms of Probability

(Kolmogorov, 1933)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Axioms of Proability

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Recall: S = all possible outcomes. E = the event.

• Axiom 1: 0 ≤ P(E) ≤ 1

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• Axiom 2: P(S) = 1

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• Axiom 3: If events E and F are mutually exclusive:

P(E ∪ F) = P(E) + P(F)

Kolmogorov

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Mutually Exclusive Events

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If events are mutually exclusive, probability of OR is simple:

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Mutually Exclusive Events

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If events are mutually exclusive, probability of OR is simple:

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

If events are mutually exclusive, the probability of OR is easy!

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Many times it is easier to calculate P(E^C)

Provable Identity #1:

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P(E^C) = 1 - P(E)

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sources of Probability:

A dice story

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Die = 7?

Roll two 6-sided dice. What is the probability that the sum = 7?

Let E be the event that the sum is 7

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Die = 7?

Roll two 6-sided dice. What is the probability that the sum = 7?

Let E be the event that the sum is 7

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Die = 7?

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cgregg % python3 dice_soln.py

After 10000000 trials

P(E) = 0.1666913

Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Die = 2?

Roll two 6-sided dice. What is the probability that the sum = 2?

Let E be the event that the sum is 2

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Die = 2?

Roll two 6-sided dice. What is the probability that the sum = 2?

Let E be the event that the sum is 2

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Other ways to make a Sample Space?

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Sum of Two Dice: Three options for the sample space

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Value of a die

Value of a die

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Sum of Two Dice: Bug: Die are Indistinct

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Dice: Three options for the sample space

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Dice: Bug: Just look at the sum

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Sum of Two Dice: Three options for the sample space

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Mehran Sahami, Chris Piech, Lisa Yan, Jerry Cain, Juliette Woodrow, and Chris Gregg, Spring 2026

Next time:

Conditional Probability

and Bayes’ Theorem

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