If $A$ and $B$ are events in $S$, and $P(B) > 0,$ then the conditional probability of $A$ given $B$, written $P(A\vert B)$, is \[P(A\vert B) = \frac{P(A\cap B)}{P(B)}.\]
Theorem 1.5.1 (Baye's Rule)
Let $A_1,A_2,\ldots$ be a partition of the sample space, and let $B$ be any set. Then for each $i=1,2,\ldots,$
\[P(A_i\vert B) = \frac{P(B \vert A_i) P(A_i)}{\sum_{j=1}^{\infty} P(B\vert A_j)P(A_j)}.\]
Two events, $A$ and $B$, are statisticaly independent if \[P(A\cap B) = P(A)P(B)\]
Theorem 1.5.2
If $A$ and $B$ are independent events, then the following pairs are also independent:
a.) $A$ and $B^c$
b.) $A^c$ and $B$
c.) $A^c$ and $B^c$
a.) $A$ and $B^c$
b.) $A^c$ and $B$
c.) $A^c$ and $B^c$
A collection of events $A_1, \ldots, A_n$ are mutually independent if for any subcollection $A_{i_1},\ldots,A_{i_k},$ we have \[P\left(\bigcap_{j=1}^{k} A_{i_j}\right) = \prod_{j=1}^{k} P(A_{i_j}).\]