
\begin{slide}
\subsection{\bf Bayes Theorem: Conditional}
\subsection{\bf Probabilities---3) Bayes Law}
\begin{itemize}
\item Now we define conditional probability $P(S/A)$ (read as probability
of $S$ given $A$ has occurred) as $P(SA)/P(A)$.  

\item Note this is a probability according to axioms, if we restrict
``bounding space'' $Y$ to be $A$.

\item Then, in terms of frequencies,
\begin{eqnarray}
P(A/B) & = & \frac{N_3}{N_2 + N_3} \hspace{1em} \mbox{as}\ N\rightarrow\infty 
\nonumber \\
P(B/A) & = & \frac{N_3}{N_1 + N_3} \hspace{1em} \mbox{as}\ N\rightarrow\infty 
\end{eqnarray}
whence
\begin{equation}
P(B) P(A/B) = P(A) P(B/A)
\end{equation}
which is {\bf Bayes Law}. 

\item Of course Equation~(3) is obvious from definition:  we didn't
need frequencies to prove it.
\end{itemize}
\end{slide}

