Elementary Properties of Probability

Building on the classical formula $P(A)=n(A)/n(S)$, several properties follow immediately and are used constantly in later sections:

1. Since $A$ and $A'$ together make up all of $S$ with no overlap, $P(A')=1-P(A)$.
2. For any event $A$, $0\le P(A)\le 1$ (a probability can never be negative or exceed 1).
3. The impossible event has probability zero: $P(\phi)=0$.
4. The certain event (the whole sample space) has probability one: $P(S)=1$.
5. If $A_1$ and $A_2$ are mutually exclusive, $P(A_1\cup A_2)=P(A_1)+P(A_2)$ - probabilities simply add when there is no overlap.
6. If $A\subseteq B$ then $P(A)\le P(B)$, and $P(A'\cap B)=P(B)-P(A)$ (the part of $B$ outside $A$).
7. Addition theorem: for any two events, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ - the overlap is subtracted once because it was counted twice. (Proved in full in section 9.2.1.)
8. For any two events, $P(A\cap B')=P(A)-P(A\cap B)$ - the part of $A$ outside $B$.
9. For three events, $P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)$.
10. If $A_1,A_2,\ldots,A_m$ are mutually exclusive, $P(A_1\cup A_2\cup\cdots\cup A_m)=P(A_1)+P(A_2)+\cdots+P(A_m)$.

Why the classical formula follows from these properties. Take a finite sample space $S=\{a_1,a_2,\ldots,a_n\}$ and let $A_i=\{a_i\}$ be the elementary event for each outcome. Since the $A_i$ partition $S$, property 10 gives $P(S)=P(A_1)+P(A_2)+\cdots+P(A_n)=1$. If every outcome is equally likely, all the $P(A_i)$ are equal, so each must be $1/n$. Now if event $A$ is made up of $m$ of these elementary outcomes, $A=A_1\cup A_2\cup\cdots\cup A_m$, property 10 again gives $P(A)=m\times(1/n)=m/n=n(A)/n(S)$ - exactly the classical formula, now derived rather than assumed. …