Probability foundations
What you’ll learn
Section titled “What you’ll learn”This is lesson 5 of Track 9 (Statistics & Probability for AI) and the opener of Phase 2 (The laws of chance). Phase 1 described data; this phase reasons about uncertainty, and it starts with the grammar. You will learn what a probability actually is, how to compute one from a sample space, and the three rules that combine probabilities, which between them handle a surprising amount of reasoning about chance. The source curriculum is Khan Academy’s Statistics & Probability course, by Sal Khan and the Khan Academy team, freely available and cited as further study.
The lesson defines probability as a number from 0 to 1 (readable as a frequency or a belief), builds the sample-space counting definition, then works the three rules on dice, coins, and cards: the complement rule with its at-least-one shortcut, the addition rule for OR (and why you subtract the overlap), and the multiplication rule for independent ANDs (and why independence is required). It closes by connecting the rules to AI: pipeline reliability, the at-least-one pattern, and how a language model scores a sentence by multiplying word-by-word probabilities.
Where this fits
Section titled “Where this fits”This is lesson 5 of 14 and the first lesson of Phase 2. It builds directly on lesson 1’s framing of AI as probabilistic. The next lesson, When one event tells you about another: conditional probability and independence, picks up exactly where this lesson’s independence caveat ends, and the lesson after that turns the base-rate example from lesson 1 into Bayes’ theorem. So this lesson is the foundation the rest of the phase is built on.
Before you start
Section titled “Before you start”Prerequisites: lesson 1 (Why AI runs on statistics) for the framing of probability as a degree of belief. Comfort with fractions and basic arithmetic is all the math you need; the examples use dice, coins, and a deck of cards.
About the math
Section titled “About the math”This lesson has the most hands-on arithmetic so far, but all of it is simple: counting outcomes, multiplying two fractions, subtracting from one. Every rule is worked on a concrete example (a die roll, two coin flips, a card draw) so the formula and the picture arrive together. There is no algebra beyond manipulating fractions.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Define probability as a number from 0 to 1 and read it as either a long-run frequency or a degree of belief
- Use the sample space and events to compute the probability of an outcome when outcomes are equally likely
- Apply the complement rule, including the at-least-one shortcut
- Apply the addition rule for OR (subtracting the overlap) and the multiplication rule for independent ANDs
- Recognize when events are not independent, so the simple multiplication rule does not apply
Time and difficulty
Section titled “Time and difficulty”- Read time: about 12 minutes
- Practice time: about 15 minutes (a self-check, a compute-the-probability exercise across dice, coins, and cards, a which-rule-and-is-it-independent exercise, and flashcards)
- Difficulty: standard (simple but real arithmetic with fractions; no algebra)