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. $P(\bar{E}\cap R)$
   3. $P(E\cap \bar{R})$
   4. $P(\bar{E}\cap \bar{R})$
   
   

   Solution

   Using the information from the text, we can directly calculate the following joint probabilities:
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{P(E\cap R)}& =\hfill & P(R|E)·P(E)=0.66·0.09=0.0594=5.94\%\hfill \\ \multicolumn{1}{c}{P(\bar{E}\cap \bar{R})}& =\hfill & P(\bar{R}|\bar{E})·P(\bar{E})=0.65·0.91=0.5915=59.15\%.\hfill \end{array}}$

   
   The remaining probabilities can then be found by calculating sums and differences in the fourfold table:

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

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