## Exam 1

1. #### 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 $E$) $74\%$ 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.)

1. $P(E \cap R)$
2. $P(\overline{E} \cap R)$
3. $P(E \cap \overline{R})$
4. $P(\overline{E} \cap \overline{R})$

#### Solution

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

1. $P(E \cap R) = 9.62\%$
2. $P(\overline{E} \cap R) = 23.49\%$
3. $P(E \cap \overline{R}) = 3.38\%$
4. $P(\overline{E} \cap \overline{R}) = 63.51\%$