Summary

The classical probability ratio. For a sample space $S$ and event $A$:

Axioms of probability. $P(A)$ satisfies: (1) $P(A)\ge0$. (2) If $A,B$ are mutually exclusive, $P(A\cup B)=P(A)+P(B)$. (3) $P(S)=1$.

Basic theorems. The probability of the impossible event is zero: $P(\varnothing)=0$. For any two events $A,B$, $P(A\cap\bar B)=P(A)-P(A\cap B)$. The Addition Theorem: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

Conditional probability. $P(B/A)=\dfrac{P(A\cap B)}{P(A)}$, provided $P(A)\ne0$; similarly $P(A/B)=\dfrac{P(A\cap B)}{P(B)}$, provided $P(B)\ne0$.

Multiplication Theorem. $P(A\cap B)=P(A/B)\,P(B)=P(B/A)\,P(A)$.

Independent events. $A,B$ are independent if and only if $P(A\cap B)=P(A)\cdot P(B)$.

Total Probability. If $A_1,A_2,\ldots,A_n$ are mutually exclusive and exhaustive and $B$ is any event in $S$, then $P(B)$ is the total probability of $B$: $P(B)=\sum_{i=1}^{n}P(A_i)\cdot P(B/A_i)$. …