Probability is the mathematics of uncertainty: a way to put a number on how likely something is. That number always lies between and : an impossible event has probability , a certain one has probability , and everything doubtful sits in between. A fair coin landing heads is ; a fair die showing a six is about .
Two readings
What does a probability actually mean? There are two respectable answers, and they usually agree.
The frequentist view says probability is long-run frequency. To say a coin has probability of heads is to say that over very many tosses, about half will be heads. The probability is a fact about repeatable trials, read off from how often things happen.
The subjective or Bayesian view says probability is a degree of belief. It lets you talk about one-off events that cannot be repeated (the chance of rain tomorrow, or that a particular theory is true) as a considered measure of confidence, updated as evidence comes in. This is the view that powers the later lesson on Bayes' theorem.
The basic rules
However you read them, probabilities obey a few simple rules. They never fall below or rise above . The probabilities of all the possible outcomes of a situation add up to exactly : something must happen. And the chance that an event does not happen is minus the chance that it does, so if rain has probability , no rain has probability .
Why it matters
Uncertainty is everywhere (in games, weather, medicine, finance, science, and daily choices) and intuition handles it badly. People overrate rare dangers, misjudge coincidences, and fall for gambler's fallacies. Probability gives a disciplined language for reasoning about chance, one that consistently beats gut feeling. The rest of this course builds that language: first counting outcomes, then combining and updating probabilities, and finally the distributions that describe whole populations of chance.