Probability

Author

Chao Ma

Published

April 8, 2026

Notes on probability foundations, from sample spaces and axioms to counting, continuous models, and random phenomena.

MIT 6.041 Probability: Discrete Random Variables III Joint PMFs for multiple random variables, multiplication rules, expectations of functions, variance of sums, binomial mean and variance, and the hat-check problem.

MIT 6.041 Probability: Discrete Random Variables II Conditional PMFs and conditional expectation, the memoryless property of the geometric distribution, total expectation, and joint PMFs.

MIT 6.041 Probability: Discrete Random Variables I Random variables as functions, probability mass functions, geometric and binomial examples, expectation as weighted average, functions of random variables, and variance.

MIT 6.041 Probability: Counting Uniform counting probabilities, permutations, subsets, binomial coefficients, binomial probabilities, and partition-based counting arguments.

MIT 6.041 Probability: Independence Independence as information-free probability, disjoint versus independent events, hidden-variable conditional independence, and why pairwise independence does not imply mutual independence.

MIT 6.041 Probability: Conditioning and Bayes’ Rule Conditional probability as belief revision, multiplication and total-probability rules, and Bayes’ rule through die and radar examples from MIT 6.041.

MIT 6.041 Probability: Probability Models and Axioms Sample spaces, discrete and continuous models, probability axioms, uniform laws, and counting examples from the opening lecture of MIT 6.041.

---
title: "Probability"
author: "Chao Ma"
date: "2026-04-08"
---

Notes on probability foundations, from sample spaces and axioms to counting, continuous models, and random phenomena.

---

::: {.content-grid}

::: {.content-card}
**[MIT 6.041 Probability: Discrete Random Variables III](discrete-random-variables-3.qmd)**
Joint PMFs for multiple random variables, multiplication rules, expectations of functions, variance of sums, binomial mean and variance, and the hat-check problem.
:::

::: {.content-card}
**[MIT 6.041 Probability: Discrete Random Variables II](discrete-random-variables-2.qmd)**
Conditional PMFs and conditional expectation, the memoryless property of the geometric distribution, total expectation, and joint PMFs.
:::

::: {.content-card}
**[MIT 6.041 Probability: Discrete Random Variables I](discrete-random-variables-1.qmd)**
Random variables as functions, probability mass functions, geometric and binomial examples, expectation as weighted average, functions of random variables, and variance.
:::

::: {.content-card}
**[MIT 6.041 Probability: Counting](counting.qmd)**
Uniform counting probabilities, permutations, subsets, binomial coefficients, binomial probabilities, and partition-based counting arguments.
:::

::: {.content-card}
**[MIT 6.041 Probability: Independence](independence.qmd)**
Independence as information-free probability, disjoint versus independent events, hidden-variable conditional independence, and why pairwise independence does not imply mutual independence.
:::

::: {.content-card}
**[MIT 6.041 Probability: Conditioning and Bayes' Rule](conditioning-bayes-rule.qmd)**
Conditional probability as belief revision, multiplication and total-probability rules, and Bayes' rule through die and radar examples from MIT 6.041.
:::

::: {.content-card}
**[MIT 6.041 Probability: Probability Models and Axioms](probability-models-axioms.qmd)**
Sample spaces, discrete and continuous models, probability axioms, uniform laws, and counting examples from the opening lecture of MIT 6.041.
:::

:::