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 $E$) $66\%$ 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.)

   
   
   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: $$\begin{aligned} P(E \cap R) & = P(R | E) \cdot P(E) = 0.66 \cdot 0.09 = 0.0594 = 5.94\%\\ P(\overline{E} \cap \overline{R}) & = P(\overline{R} | \overline{E}) \cdot P(\overline{E}) = 0.65 \cdot 0.91 = 0.5915 = 59.15\%. \end{aligned}$$ The remaining probabilities can then be found by calculating sums and differences in the fourfold table:

   	$R$	$\overline{R}$	sum
   $E$	5.94	3.06	9.00
   $\overline{E}$	31.85	59.15	91.00
   sum	37.79	62.21	100.00

   
   
   1. $P(E \cap R) = 5.94\%$
   2. $P(\overline{E} \cap R) = 31.85\%$
   3. $P(E \cap \overline{R}) = 3.06\%$
   4. $P(\overline{E} \cap \overline{R}) = 59.15\%$