LLM Primer

Chapter 7

Probability

Events, random variables, distributions, expectation, variance, conditional probability, and Bayes' rule.

What this chapter does

Probability gives uncertainty a precise language. This chapter moves from events to random variables and distributions, then uses expectation, variance, conditioning, and Bayes' rule to reason about data and model outputs.

Lessons

Read these in order.

Start with the chapter introduction, then move through the topic lessons. The order is chosen so each page can reuse ideas from the pages before it.

  1. 01
    Introduction

    Why probability is the language of uncertainty in machine learning.

  2. 02
    Sample Spaces and Events

    Outcomes, events, and probability as a measure of event size.

  3. 03
    Probability Rules

    Complements, addition, overlap, and consistency rules.

  4. 04
    Random Variables

    Turning uncertain outcomes into quantities we can compute with.

  5. 05
    Distributions

    How probability is assigned to possible values.

  6. 06
    Expectation

    Probability-weighted averages and expected loss.

  7. 07
    Variance and Covariance

    Spread, standard deviation, and variables moving together.

  8. 08
    Independence

    When one event does not change another event's probability.

  9. 09
    Conditional Probability

    Probability after context or evidence is known.

  10. 10
    Bayes' Rule

    Updating probability after evidence.

  11. 11
    Common Distributions

    Bernoulli, categorical, multinomial, and Gaussian distributions.

Review and practice

Close the chapter deliberately.

Use the conclusion and revision notes before the chapter exercises. Hints and solutions are collected here, while lesson-level exercises reveal their own help inline.

Conclusion

What Chapter 7 accomplished and how probability prepares numerical computation.

Summary and Revision Notes

A compact review of probability rules, distributions, expectation, and Bayes' rule.

Exercises

Chapter-level practice for events, distributions, expectation, evidence, and Bayes' rule.

Hints

Low-spoiler nudges for the Chapter 7 exercises.

Solutions

Explained solutions for the Chapter 7 exercises.

Before moving on

  • Interpret model outputs probabilistically.
  • Use distributions to reason about data and uncertainty.
  • Distinguish events, random variables, and distributions.
  • Read conditional probability and Bayes' rule as update language.

Where this leads

  • Statistics
  • Information Theory

Chapter progress