Probability Fundamentals

Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain). The sample space $U$ is the set of all possible outcomes of an experiment, and an event is any subset of that sample space. When outcomes are equally likely, the probability of event $A$ is $P(A) = \dfrac{n(A)}{n(U)}$, the number of favorable outcomes divided by the total number of outcomes. The complement $A'$ is the event that $A$ does not happen, and since something must occur, $P(A) + P(A') = 1$, giving the very useful $P(A') = 1 - P(A)$.

Combined events

For two events $A$ and $B$, the union $A \cup B$ means "$A$ or $B$ (or both)" and the intersection $A \cap B$ means "$A$ and $B$ together." These are linked by the addition rule: $$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$ We subtract the i