

\begin{slide}
\subsection{\bf Example: Analysis of Unreliable Test}
\begin{itemize}
\item Consider (page 18, Yost) set $B$ to categorize health of people
and set $A$ to be results of unreliable test.

\item $B$ has two possible results.
\begin{eqnarray*}
B_1 = 0 & \hspace{1em} & \mbox{ill} \\
B_2 = 1 & \hspace{1em} & \mbox{well}
\end{eqnarray*}

\item Consider the situation with $A = $ ``Test Positive''

\item Bayes gives for $B=B_1$ or $B_2$

\[
P(B/A) = \frac{P(B) P(A/B)}{P(A)}
\]

\item Apply separately $B=B_1$, or $B_2$,\ i.e., 

$B=B_1 + B_2$ and $P(B/A) = P(B_1/A) + P(B_2/A)$.

\item Then given data proposed by Yost
\begin{eqnarray*}
&&P\left(B_2\right)\ \ P\left(A/B_2\right)\ \ \ P\left(B_1\right)
\ P\left(A/B_1\right) \\
P(B/A) & = & \frac{\lbrack .999 \times 0.02\delta (1-x) + .001 \times 
.99\delta (x) \rbrack}{\lbrack P(A) = .001 \times .99 + .02 \times
.999 \rbrack} \\
&& \\
P\left(B_1/A\right) & = & \mbox{``ill''} = .047 \\
&& \\
P\left(B_2/A\right) & = & \mbox{``well''} = .953
\end{eqnarray*}
\end{itemize}
\end{slide}

