In exams: Automatic Generation of Exams in R

ok <- FALSE
while(!ok) {
  pe <- round(runif(1, 0.05, 0.15), digits = 2)
  per <- round(runif(1, 0.6, 0.8), digits = 2)
  pnenr <- round(runif(1, 0.6, 0.8), digits = 2)

  prob1 <- pe * per
  prob2 <- pe * (1 - per)
  prob3 <- (1 - pe) * (1 - pnenr)
  prob4 <- (1 - pe) * pnenr

  tab <- cbind(c(prob1, prob3), c(prob2, prob4))

  sol <- c(tab[1, 1] / sum(tab[, 1]), tab[1, 1] / sum(tab[1, ]),
    tab[2, 1] / sum(tab[2, ]), tab[1, 2] / sum(tab[1, ]))

  ok <- sum(tab) == 1 & all(tab > 0) & all(tab < 1)
}

tab2 <- cbind(rbind(tab, colSums(tab)), c(rowSums(tab), 1))
tab2 <- format(tab2 * 100, digits = 2, nsmall = 2)
tab2 <- gsub(" ", "", tab2, fixed = TRUE)

sol <- round(100 * c(tab, colSums(tab), rowSums(tab), sum(tab)), digits = 2)
lab <- c("E \\cap R", "\\overline{E} \\cap R", "E \\cap \\overline{R}", "\\overline{E} \\cap \\overline{R}",
  "R", "\\overline{R}", "E", "\\overline{E}", "\\Omega")

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 $E$) $r per * 100\%$ get a positive recommendation (event $R$). However, out of those candidates that are not eligible $r pnenr * 100\%$ get a negative recommendation. Overall, they know that only $r pe * 100\%$ of all job applicants are actually eligible.

What is the corresponding fourfold table of the joint probabilities? (Specify all entries in percent.)

| | $R$ | $\overline{R}$ | sum | |:-------------:|:-------------:|:--------------:|:-------------:| |$E$ | ##ANSWER1##\% | ##ANSWER3##\% | ##ANSWER7##\% | |$\overline{E}$ | ##ANSWER2##\% | ##ANSWER4##\% | ##ANSWER8##\% | |sum | ##ANSWER5##\% | ##ANSWER6##\% | ##ANSWER9##\% |

answerlist(rep("", length(sol)), markup = "markdown")

Solution

Using the information from the text, we can directly calculate the following joint probabilities: $$ \begin{aligned} P(E \cap R) & = P(R | E) \cdot P(E) = r per \cdot r pe = r prob1 = r 100 * prob1\%\ P(\overline{E} \cap \overline{R}) & = P(\overline{R} | \overline{E}) \cdot P(\overline{E}) = r pnenr \cdot r 1 - pe = r prob4 = r 100 * prob4\%. \end{aligned} $$ The remaining probabilities can then be found by calculating sums and differences in the fourfold table:

| | $R$ | $\overline{R}$ | sum | |:-------------:|:------------------:|:------------------:|:------------------:| |$E$ | r tab2[1, 1] | r tab2[1, 2] | r tab2[1, 3] | |$\overline{E}$ | r tab2[2, 1] | r tab2[2, 2] | r tab2[2, 3] | |sum | r tab2[3, 1] | r tab2[3, 2] | r tab2[3, 3] |

answerlist(paste("$P(", lab, ") = ", format(sol), "\\%$", sep = ""), markup = "markdown")

Meta-information

extype: cloze exsolution: r paste(sol, collapse = "|") exclozetype: num|num|num|num|num|num|num|num|num exname: fourfold extol: 0.05 exextra[numwidth,logical]: TRUE