

\begin{question}
An industry-leading company seeks a qualified candidate for a management position.
A management consultancy carries out an assessment center which concludes in making
a positive or negative recommendation for each candidate: From previous assessments they know that
of those candidates that are actually eligible for the position (event $E$) $66\%$
get a positive recommendation (event $R$). However, out of those candidates that are not eligible
$65\%$ get a negative recommendation. Overall, they know that only
$9\%$ of all job applicants are actually eligible.

What is the corresponding fourfold table of the joint probabilities? (Specify all entries in percent.)

\begin{answerlist}
  \item $P(E \cap R)$
  \item $P(\overline{E} \cap R)$
  \item $P(E \cap \overline{R})$
  \item $P(\overline{E} \cap \overline{R})$
\end{answerlist}
\end{question}


\begin{solution}
Using the information from the text, we can directly calculate the following joint probabilities:
\begin{eqnarray*}
  P(E \cap R) & = &
    P(R | E) \cdot P(E) = 0.66 \cdot 0.09 = 0.0594 = 5.94\%\\
  P(\overline{E} \cap \overline{R}) & = &
    P(\overline{R} | \overline{E}) \cdot P(\overline{E}) = 0.65 \cdot 0.91 = 0.5915 = 59.15\%.
\end{eqnarray*}
The remaining probabilities can then be found by calculating sums and differences in the fourfold table:

\begin{tabular}{|c|cc|c|}
\hline
                 & ~$R$~                    & ~$\overline{R}$~         & sum \\ \hline
~$E$~            & ~{\bf   5.94}~ & ~{\it   3.06}~ & ~{\bf   9.00}~ \\
~$\overline{E}$~ & ~{\it  31.85}~ & ~{\bf  59.15}~ & ~{\it  91.00}~ \\ \hline
sum              & ~{\it  37.79}~ & ~{\it  62.21}~ & ~{\bf 100.00}~ \\
\hline
\end{tabular}

\begin{answerlist}
  \item $P(E \cap R) =  5.94\%$
  \item $P(\overline{E} \cap R) = 31.85\%$
  \item $P(E \cap \overline{R}) =  3.06\%$
  \item $P(\overline{E} \cap \overline{R}) = 59.15\%$
\end{answerlist}
\end{solution}


%% META-INFORMATION
%% \extype{cloze}
%% \exsolution{5.94|31.85|3.06|59.15}
%% \exclozetype{num|num|num|num}
%% \exname{fourfold}
%% \extol{0.05}
