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 EE) 74%74\% get a positive recommendation (event RR). However, out of those candidates that are not eligible 73%73\% get a negative recommendation. Overall, they know that only 13%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(ER)P(E \cap R)
    2. P(E¯R)P(\overline{E} \cap R)
    3. P(ER¯)P(E \cap \overline{R})
    4. P(E¯R¯)P(\overline{E} \cap \overline{R})

    Solution

    Using the information from the text, we can directly calculate the following joint probabilities: P(ER)=P(R|E)P(E)=0.740.13=0.0962=9.62%P(E¯R¯)=P(R¯|E¯)P(E¯)=0.730.87=0.6351=63.51%. \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:

    RR R¯\overline{R} sum
    EE 9.62 3.38 13.00
    E¯\overline{E} 23.49 63.51 87.00
    sum 33.11 66.89 100.00

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