Probability rules explained
Probability has a small set of rules for combining the chances of events: the complement, addition, and multiplication rules. This guide explains each one with examples so you can handle and, or, and not questions.
The basics
A probability is a number from 0 to 1 measuring how likely an event is, where 0 is impossible and 1 is certain. For equally likely outcomes, the probability of an event is favorable outcomes over total outcomes. The rules below combine such probabilities for more complex situations.
The complement rule
The probability that an event does not happen is 1 minus the probability that it does. If the chance of rain is 0.3, the chance of no rain is 0.7. This rule is handy when the opposite event is easier to count than the event itself.
The addition rule (or)
For the probability that A or B happens, add their probabilities and subtract the overlap: P(A) plus P(B) minus P(A and B). The subtraction prevents double-counting outcomes that are in both events. When A and B cannot both happen, they are mutually exclusive, the overlap is zero, and you simply add.
The multiplication rule (and)
For the probability that A and B both happen, multiply: P(A) times P(B) for independent events, where one does not affect the other. If the events are dependent, use the conditional probability of B given A. Drawing two cards without replacement is dependent; flipping two coins is independent.
Two coin flips both landing heads are independent, so P = 0.5 × 0.5 = 0.25.
- Probability runs from 0 (impossible) to 1 (certain).
- Complement: P(not A) = 1 − P(A).
- Addition (or): P(A) + P(B) − P(A and B).
- Multiplication (and): P(A) × P(B) for independent events.
- Mutually exclusive events have no overlap to subtract.
Related tools
Use the Probability Calculator, or see the Probability Formula.
FAQ
What is the complement rule?
The probability an event does not occur is 1 minus the probability it does.
When do I add probabilities versus multiply them?
Add (with the overlap removed) for or questions; multiply for and questions with independent events.
What are mutually exclusive events?
Events that cannot both happen at once, so their overlap probability is zero and you simply add them.