
\problemdata     
{S01-PS9-4}           
{probability-conditional}              
{} 		
{S01-PS9}
{S01 Karger}



\begin{problem}



Each person carries a certain, rare disease
with probability 1 in 1000.  Two doctors claim to be able to test for
the disease before symptoms appear.

\begin{problemparts}

\problempart Doctor $A$ was educated at Harvard Medical School and
employs the latest high-tech test.  If a patient has the disease, then
the test comes back positive 99\% of the time.  If the patient does
not have the disease, then the test comes back negative 97\% of the
time.  What is the probability that Doctor $A$ correctly diagnoses a
patient?

\solution{
\begin{eqnarray*}
\Pr(\text{$A$ is correct})
    & = &   \Pr(\text{is diseased} \cap \text{test positive}) \\
    &   &   \mbox{} +
        \Pr(\text{not diseased} \cap \text{test negative}) \\
    & = &   \Pr(\text{is diseased}) \cdot
        \Pr(\text{test positive} \mid \text{is diseased}) \\
    &   &   \mbox{} +
        \Pr(\text{not diseased}) \cdot
        \Pr(\text{test negative} \mid \text{not diseased}) \\
    & = &   0.001 \cdot 0.99 + 0.999 \cdot 0.97 \\
    & \approx & 0.970
\end{eqnarray*}
}

\problempart Doctor $B$ (a quack from MIT) murmurs strange chants and burns incense.
With probability 1 in 1000, Doctor $B$ says that a patient has the
disease, regardless of whether or not the patient actually does.  What
is the probability that Doctor $B$ correctly disgnoses a patient?

\solution{
\begin{eqnarray*}
\Pr(\text{$B$ is correct})
    & = &   \Pr(\text{is diseased} \cap \text{test positive}) \\
    &   &   \mbox{} +
        \Pr(\text{not diseased} \cap \text{test negative}) \\
    & = &   \Pr(\text{is diseased}) \cdot
        \Pr(\text{test positive} \mid \text{is diseased}) \\
    &   &   \mbox{} +
        \Pr(\text{not diseased}) \cdot
        \Pr(\text{test negative} \mid \text{not diseased}) \\
    & = &   0.001 \cdot 0.001 + 0.999 \cdot 0.999 \\
    & \approx & 0.998
\end{eqnarray*}
}

\problempart Which doctor makes more errors? And how often, 
compared to the other?

\solution{

\textbf{Solution:}\\
Doctor $A$ makes about 15 times more errors than Doctor
$B$.}

\eparts
\end{problem}