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\begin{document}

\recn{10}{4/17}{April 17, 1998}


\begin{problems}

\problem Verify that the set-theoretic probability equations above
``relativize'' to conditional probabilities, e.g., equation~(1)
relativizes to
\[\pr{A-B\mid C}  =  \pr{A \mid C} - \pr{A \intersect B \mid C}.\]


\problem
In the OJ Simpson trial it was claimed by the defense that the fact that
Simpson battered his wife should be excluded from the evidence as the fraction
of wife batterers that murder their wives is $1/2000$ (Dershowitz Lemma).
In this problem we will condition this probability on the event that Nicole is 
dead and see that we obtain a somewhat different result.

We have the following statistical information (all for 1994):
\begin{itemize}

\item number of women = 125M
\item number of women murdered in 1994 = 6000
\item number of women murdered by spouse in 1994 = 1500
\item number of women with spouse in 1994 = 50M
\item number of women battered by spouse in 1994 = 3M
\end{itemize}

Being murdered counts as being battered. Also, we assume that 
the event of being murdered by
somebody else is not influenced by whether the woman was being battered or not.

Define the events

$S$ = Woman is murdered by spouse  \\
$M$ = Woman is murdered \\
$B$ = Woman is battered by spouse \\

Use the given information to evaluate the following

\bparts

\ppart  The Dershowitz Lemma can be reformulated as
\[\pr{S \mid B} = 1/2000.\]
Prove it.

\ppart Calculate $\pr{S \mid M}$, i.e., the probability that a murdered
woman was done in by her spouse.

\ppart Calculate $\pr{S \mid B \intersect M}$, i.e., the probability that
a battered, murdered woman is done in by her spouse.

\ppart What can you conclude about Dershowitz' argument that the battery
of a spouse should not be admitted as evidence because of his lemma ?

\eparts

\end{problems}

\end{document}

