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\tutorial{tutorial8}{Problems for Tutorial 8}{November 18, 1996}

\section {Some Useful Identities}

\begin{enumerate}

\item Let $A_1, \ldots , A_n$ be a collection of disjoint sets such that their 
union is equal to the sample space.  Prove by induction that for any event $B$

$P(B) = P(B \cap A_1)+ \ldots +P(B \cap A_n)$

\item Use part 1 to show that if $A_1, \ldots ,A_n$ is a disjoint collection
that covers the sample space, then for any events $B,C$ such that $P(C) \neq 0$

$P(B \mid C) = P(B \cap A_1 \mid C) + ... +P(B \cap A_n \mid C)$

Observe the useful corollary:

$P(B \mid C) = P(B \cap A \mid C) + P(B \cap \bar{A} \mid C)$

\end{enumerate}

\section{Trial By Jury}


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 Nicol is 
dead and see that we obtain a somewhat different result.

We have the following statistical information:

\begin{itemize}

\item number of women = 125M
\item number of women murdered in 1994 = 6000
\item number of women murdered by souse 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, the event of being murdered by
somebody else is not influenced by whether the woman was being battered or not.

\newpage 
Define the events

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

Use the given information to evaluate the following

\begin{itemize}

\item  Dershowitz Lemma: $Pr(G \mid B) = 1/2000$
\item  $Pr(G \mid B \cap M)$
\item  $Pr(G \mid M)$

\end{itemize}

Do you think that the evidence that Simpson battered his wife should have been
excluded based on Dershowitz Lemma?



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