## 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. $P\left(E\cap R\right)$
2. $P\left(\stackrel{‾}{E}\cap R\right)$
3. $P\left(E\cap \stackrel{‾}{R}\right)$
4. $P\left(\stackrel{‾}{E}\cap \stackrel{‾}{R}\right)$

#### Solution

Using the information from the text, we can directly calculate the following joint probabilities:
 $\begin{array}{ccc}\multicolumn{1}{c}{P\left(E\cap R\right)}& =\hfill & P\left(R|E\right)·P\left(E\right)=0.66·0.09=0.0594=5.94%\hfill \\ \multicolumn{1}{c}{P\left(\stackrel{‾}{E}\cap \stackrel{‾}{R}\right)}& =\hfill & P\left(\stackrel{‾}{R}|\stackrel{‾}{E}\right)·P\left(\stackrel{‾}{E}\right)=0.65·0.91=0.5915=59.15%.\hfill \end{array}$

The remaining probabilities can then be found by calculating sums and differences in the fourfold table:
 $R$ $\stackrel{‾}{R}$ sum $E$ 5.94 3.06 9.00 $\stackrel{‾}{E}$ 31.85 59.15 91.00 sum 37.79 62.21 100.00

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