Exercise 4 · Probability and matrices — University students (Option A)
Simple and conditional probability on a contingency table; matrix products for totals and revenue.
Maximum score · 2.5 pointsThe following table shows the number of bachelor (G), master's (M) and PhD (D) students enrolled in the three faculties —Science (C), Engineering (I) and Humanities (L)— of a small university.
- If a student is chosen at random, what is the probability that they are a PhD student from the Faculty of Humanities? If a master's student is chosen at random, what is the probability that they study at the Faculty of Engineering? 1.25 p
- Express the information in the table (without the totals) as a $3 \times 3$ matrix and compute, using a matrix product, the vector with the total number of bachelor, master's and PhD students at the university. Bachelor students pay 1000 € in fees; master's students, 1500 €; and PhD students, 500 €. Using a matrix product, compute the total fee revenue of the university. 1.25 p
Step-by-step solution
Key idea: in a contingency table, the probability of an intersection is the cell divided by the grand total, and a conditional probability is the cell divided by the total of the conditioning row or column.
a) Simple and conditional probability
Probability of being a PhD student and from Humanities (cell $D \cap L$ over the grand total):
Given that the student does a master's, the conditioning set is column M (533 students):
b) Matrix products
The $3 \times 3$ matrix of the table (rows = faculties, columns = levels G, M, D):
To add the columns (totals per level), multiply on the left by the row vector $(1\;\;1\;\;1)$:
That is, 1333 bachelor students, 533 master's and 134 PhD.
With the fee vector $(1000,\ 1500,\ 500)$ as a column: