Probability

The language of probability

Probability quantifies uncertainty on a scale from $0$ (impossible) to $1$ (certain). An experiment is a process with uncertain outcomes; the sample space $U$ is the set of all possible outcomes; an event is a subset of the sample space. For equally likely outcomes, $$P(A) = \frac{n(A)}{n(U)} = \frac{\text{number of outcomes in } A}{\text{total number of outcomes}}.$$ Rolling a fair six-sided die, the sample space is ${1,2,3,4,5,6}$ and $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$. The complement $A'$ (not $A$) satisfies $P(A') = 1 - P(A)$, often the fastest route to an answer.

Combined events

For two events $A$ and $B$, the union $A \cup B$ ("$A$ or $B$") and intersection $A \cap B$ ("$A$ and $B$") are linked by the addition rule: $$P(A \cup B) = P(A) + P(B) -