%spring99 final problem 7


\comment{Tina}{Problem seems okay.  Did we emphasize this in class?  Part
c will be easy if they remember it from notes, etc but if not, it may take
a little while to find an example.}

\begin{problem}[10]

We consider a hypothetical Institute with only two departments: EECS
and Math, and consider the experiment where we pick a random applicant
for admission to one of these departments.  Define the following
events:
\begin{center}
\begin{tabular}{rcl}
A & ::= & applicant is accepted,\\
F & ::= & applicant is female,\\
M & ::= &  applicant is male,\\
MATH & ::= & applicant applied to Math,\\
EECS & ::= & applicant applied to EECS.
\end{tabular}
\end{center}
Assume that all applicants are either male or female, and that no
applicant applied to both departments.
\iffalse
That is, the events $F_{EECS}$,
$F_{CS}$, $\mbox{M-EECS}$, and $\mbox{M-CS}$ are all disjoint.
\fi

\begin{problemparts}

\problempart[5] The Institute is sued for gender discrimination, with
the plaintiff observing that in both EECS and Math, the probability that a
woman is accepted is less than the probability that a man is accepted.
This observation can be precisely expressed with two inequalities
between conditional probabilities involving the events above.  State these
inequalities.

\solution[\vspace*{5cm}]{
\begin{eqnarray*}
\Pr(A \mid F \intersect EECS) & < & \Pr(A \mid M \intersect EECS), \\
\Pr(A \mid F \intersect MATH) & < & \Pr(A \mid M \intersect MATH).
\end{eqnarray*}
}

\problempart[5] The Institute's defense attorneys retort that overall among all
applicants, a woman is {\em more} likely to be accepted than a man.
Express this observation as an inequality between conditional probabilities
involving the events above.

\solution[\vspace*{4cm}]{
\begin{eqnarray*}
\prob{A \mid F} & > & \prob{A \mid M}
\end{eqnarray*}
}

\iffalse
\problempart Although the assertions above may appear to be logically
inconsistent, both the plaintiff's and the defense's claims about the
conditional probabilities can be correct. (Never mind whose interpretation
of the data we find more persuasive.)  Fill in the missing table entries
below with possible applicant statistics for which both the plaintiff and
the University attorneys' assertions are true.

\begin{center}
\renewcommand{\arraystretch}{.3}
\begin{tabular}{|c|c|c|c|}
\hline
& &\hspace*{1in}&\\
     & Applied & Accepted & percent accepted\\
 & & & \\ \hline \hline & & &\\
Dept & Females & Females &  Females\\
& & & \\ \hline & & &\\
Math & & & \\ & & & \\ \hline & & &\\
EECS & 100 & 70 & 70\% \\ & & & \\ \hline & & &\\
Total & &  & \\ & & &\\ \hline \hline & & &\\
Dept & Males & Males & Males\\
& & & \\ \hline & & &\\
Math & 100 & 50 & 50\% \\ & & & \\ \hline & & &\\
EECS & & & \\ & & & \\ \hline & & &\\
Total & &  & \\
& & &\\
\hline
\end{tabular}
\end{center}

\solution{
\[
\begin{array}{ccr}
\mbox{CS} & \mbox{0 females accepted, 1 applied} & 0\% \\
          & \mbox{50 males accepted, 100 applied} & 50\% \\
\mbox{EECS} & \mbox{70 females accepted, 100 applied} & 70\% \\
          & \mbox{1 male accepted, 1 applied} & 100\% \\
\hline
\mbox{Overall} & \mbox{70 females accepted, 101 applied} & \approx 70\% \\
          & \mbox{51 males accepted, 101 applied} & \approx 51\%
\end{array}
\]
}
\fi
\end{problemparts}
\end{problem}
