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 $E$) $74\%$ get a positive recommendation (event $R$). However, out of those candidates that are not eligible $73\%$ get a negative recommendation. Overall, they know that only $13\%$ of all job applicants are actually eligible.
What is the corresponding fourfold table of the joint probabilities? (Specify all entries in percent.)
$R$ | $\overline{R}$ | sum | |
---|---|---|---|
$E$ | % | % | % |
$\overline{E}$ | % | % | % |
sum | % | % | % |
Using the information from the text, we can directly calculate the following joint probabilities: $\begin{aligned} P(E \cap R) & = P(R | E) \cdot P(E) = 0.74 \cdot 0.13 = 0.0962 = 9.62\%\\ P(\overline{E} \cap \overline{R}) & = P(\overline{R} | \overline{E}) \cdot P(\overline{E}) = 0.73 \cdot 0.87 = 0.6351 = 63.51\%. \end{aligned}$ The remaining probabilities can then be found by calculating sums and differences in the fourfold table:
$R$ | $\overline{R}$ | sum | |
---|---|---|---|
$E$ | 9.62 | 3.38 | 13.00 |
$\overline{E}$ | 23.49 | 63.51 | 87.00 |
sum | 33.11 | 66.89 | 100.00 |