Summary

Summary

This chapter built probability on a rigorous, set-theoretic foundation.

• A random experiment has more than one possible outcome, none of which can be predicted in advance; a single performance of it is a trial, and its result is an outcome.
• The sample space $S$ is the set of all possible outcomes; each element is a sample point. $S$ can be written in roster form (e.g. $S = \{H, T\}$) or set-builder form (e.g. $S = \{(x,y) : x, y \in \{1,\dots,6\}\}$ for two dice, giving $n(S) = 36$).
• An event is any subset $E \subseteq S$. $S$ itself is the sure event; $\phi$ is the impossible event. A one-point subset is a simple event; a larger subset is a compound event.
• Set operations translate directly into event language: "not $A$" is $A'$, "$A$ and $B$" is $A \cap B$, "$A$ or $B$" is $A \cup B$.
• Events $A, B$ are mutually exclusive if $A \cap B = \phi$; a collection of events is exhaustive if their union is $S$ — these are independent properties.
• The axiomatic approach defines a probability function $P$ by three axioms: $P(E) \ge 0$; $P(S) = 1$; and for mutually exclusive $E, F$: $P(E \cup F) = P(E) + P(F)$. For equally likely outcomes this reduces to $P(E) = \dfrac{n(E)}{n(S)}$.
• Complement rule (derived): $P(A') = 1 - P(A)$, since $A \cup A' = S$ and $A \cap A' = \phi$. …