Probability rules

The complement, addition and multiplication rules combine probabilities of events; all follow from probabilities being between 0 and 1 and summing to 1 over the sample space.

P(A^c) = 1 - P(A)

Complement rule.

$A^c$: The event 'not A'.

P(A \cup B) = P(A) + P(B) - P(A \cap B)

General addition rule (for mutually exclusive events the last term is 0).

$A \cup B$: A or B (or both).
$A \cap B$: A and B.

P(A \cap B) = P(A)\,P(B \given A)

General multiplication rule; equals P(A)P(B) when A and B are independent.

$P(B \given A)$: Probability of B given that A occurred.

P(B) = \sum_{k} P(B \given A_k)\,P(A_k)

Law of total probability (the A_k partition the sample space).

$A_1, A_2, \ldots$: Mutually exclusive events that together cover every outcome.

When to use

Combining probabilities of simple events into probabilities of compound events.
Probability trees: multiply along branches, add across branches leading to the same outcome.
"At least one" questions: $P(\text{at least one}) = 1 - P(\text{none})$ .

Assumptions

Equally likely outcomes only when stated (fair dice, random draws); otherwise probabilities come from a model or long-run relative frequencies.

How to interpret

Every probability is between 0 and 1. The probabilities of all outcomes in the sample space sum to 1. Mutually exclusive events cannot happen together; independent events do not influence each other — these are different properties.

Common mistakes

Adding probabilities of events that overlap without subtracting the intersection.
Multiplying probabilities of dependent events as if independent.
Treating "mutually exclusive" as "independent" (two exclusive events with positive probabilities are always dependent).

Example

$P(A) = 0.5$ , $P(B) = 0.4$ , $P(A \cap B) = 0.2$ : $P(A \cup B) = 0.5 + 0.4 - 0.2 = 0.7$ and $P(\text{neither}) = 0.3$ . Since $P(A)P(B) = 0.2 = P(A \cap B)$ , A and B happen to be independent.