What probability measures

Probability is how we put a number on uncertainty. That number always lives between 0 (impossible) and 1 (certain), and for equally likely outcomes it comes from a simple count.

Sample space and events

Two words carry the whole idea:

• The sample space is the set of all possible outcomes. For one die roll, it's 6.
• An event is any outcome or group of outcomes you care about — like "roll an even number," 6.

When every outcome is equally likely, probability is just favorable outcomes divided by total outcomes:

So P(even) = 3/6 = 1/2. Count the slots that count, divide by all the slots.

The rules every probability obeys

• Every probability is between 0 and 1: 0 ≤ P(A) ≤ 1.
• The probabilities of all outcomes in the sample space add to 1.
• The complement rule: the chance an event does not happen is P(not A) = 1 − P(A). (Often the easy way in: "at least one" is usually best found as 1 − "none.")

Probability isn't a guarantee

A probability of 1/2 doesn't mean every other flip is heads — it means that over many flips the fraction of heads settles near one half. This law of large numbers is why casinos and insurers profit from steady odds even though any single bet is uncertain. Short runs are streaky; the long run is reliable.

Why this matters

Every risk, forecast, and game of chance rests on this counting idea. Get fluent with sample spaces and the complement rule and you can already answer a surprising range of questions — and spot when a "1 in a million" claim doesn't add up.