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) 66%66\% get a positive recommendation (event RR). However, out of those candidates that are not eligible 65%65\% get a negative recommendation. Overall, they know that only 9%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(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)=0.660.09=0.0594=5.94%P(E¯R¯)=P(R¯|E¯)P(E¯)=0.650.91=0.5915=59.15%. \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:

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

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