Probability Basics

Probability is the mathematical language of uncertainty. Every time you check a weather forecast, evaluate a medical test result, or decide whether to buy insurance, you are — whether you realize it or not — reasoning about probabilities. Statistics and probability are inseparable: while descriptive statistics summarize what has already been observed, probability provides the framework for predicting what has not yet happened and for quantifying how confident we should be in our predictions.

At its core, the probability of an event is simply the ratio of favorable outcomes to total possible outcomes, provided every outcome is equally likely. Roll a fair die and the probability of landing on a 4 is one out of six. Draw a card from a standard deck and the probability of getting a heart is thirteen out of fifty-two. These simple fractions are the building blocks of a much richer theory that can handle compound events, conditional information, and independence.

Two fundamental rules extend this foundation to more complex situations. The complement rule tells you that the probability of an event not happening is one minus the probability that it does happen — a wonderfully simple shortcut when counting "not" outcomes directly would be tedious. The addition rule lets you compute the probability that at least one of two events occurs, carefully subtracting any overlap so that shared outcomes are not counted twice. And when two events have no influence on each other — when they are independent — the probability that both occur is simply the product of their individual probabilities.

In this lesson you will practice computing probabilities from equally-likely outcome models, applying the complement and addition rules, and recognizing when events are independent. These skills provide the essential toolkit for the probability distributions and inference methods you will encounter in the lessons ahead.