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%==================== END PREAMBLE ==============
\begin{document}
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\vspace*{-80pt}

{\bf  Math 210, Spring 2019\hfill Jerry L. Kazdan}

\Large

\begin{center}
{\LARGE\bf Conditional Probability}
\smallskip
\end{center}

Notation:  $P(A|B)$ means the probability that $A$ is true assuming
that $B$ is true.

Two identities:
$$
P(A\cap B)=P(A|B)P(B) \qquad P(B\cap A)=P(B|A)P(A)
$$
Since $A\cap B = B\cap A$,
$$
P(A|B)=\frac{P(B|A)P(A)}{P(B)} \qquad \text{[Bayes]}
$$
\bigskip

{\bf Dart Board}  The dart board is the unit disk, $D=\{x^2+y^2\le 1\}$.
Suppose you know the dart landed in the upper half, $y>0$.  What is
the probability thar it landed within a distance of $1/2$ from the
origin?
\smallskip

Let $E=\{(x,y)\in D\,:\, y>0\}$ and $F=\{(x,y)\in D\,:\, x^2+y^2 <(1/2)^2\}$.
Then
$$
\begin{aligned}
P(F|E)=&\frac{P(F\cap E)}{P(E)}=\frac{(1/\pi)(1/2)(\pi/4)}{(1/\pi)(\pi/2)}\\
 =& 1/4
\end{aligned}
$$
Note that the size of $F\cap E$ is $1/4$ the size of $E$.

\bigskip

{\bf Cancer test.}  Say you test positive.  What is the likelihood you
have cancer?
$$
\begin{aligned}
P(C|+) =& \frac{P(+|C)P(C)}{P(+)} 
=&\frac{P(+|C)P(C)}{P(+|C)P(C) + P(+|-C)P(-C)}
\end{aligned}
$$
\bigskip

{\sc Remark:}

{\parindent 25pt
$P(+|-C)P(-C)$ is the probability of \emph{False Positives}
\smallskip

$P(-|C)P(C)$ is the probability of \emph{False Negatives}
}
\bigskip

We look at an example with real data.

\vfill
%[Last revised: \today]
\end{document}
