Probability Fundamentals

The one idea: Two events can be "independent" (one doesn't affect the other) or "mutually exclusive" (they can't both happen) — and these are opposite properties, not synonyms. Which addition or multiplication rule you're allowed to use depends entirely on which one you've got, and mixing them up produces confidently wrong probabilities.

⚠️ The trap: "The launch failing and the launch getting delayed are independent risks — one doesn't affect the other, so I can just multiply their probabilities." That feels right and it's wrong. Two events with any real-world overlap or exclusion between them are almost never independent — a delayed launch changes the probability of a failure (fixing bugs, but also rushing at the end), so check whether one event's probability actually shifts when you condition on the other before you multiply anything.

Two events. Two probabilities. Combine them into one number — and there are exactly two ways to combine probabilities (add them, or multiply them), and exactly two special relationships between events that make the easy versions of those rules legal (mutually exclusive, for addition; independent, for multiplication). Everyone learns the four rules. Almost everyone, at some point, applies the easy version of a rule to a situation that doesn't qualify — because the two special relationships sound like they should be the same thing, and they are, in fact, close to opposites.

This lesson builds probability from the ground up: what a probability is, the addition and multiplication laws, and the one distinction — independent vs. mutually exclusive — that causes more downstream errors than any other idea in this unit.

What a probability actually is

A probability is…