Probability Basics

Sample spaces and events

The sample space $\Omega$ is the set of all possible outcomes. An event is a subset $A \subseteq \Omega$.

Axioms of probability

1. $P(A) \ge 0$ for every event $A$.
2. $P(\Omega) = 1$.
3. If $A_1, A_2, \dots$ are disjoint, $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.

Conditional probability and Bayes’ rule

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}.$$