Exercise 4 · Probability and scholarship holders (Option B)
Simple and conditional probability on a contingency table; a linear system to find scholarship holders per level.
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 master's student from the Faculty of Science? If a student from the Faculty of Engineering is chosen at random, what is the probability that they are doing a master's? 1.25 p
- It is known that, in total, 1036 students hold a scholarship. It is also known that, taking bachelor and master's students together, half of them hold scholarships, and that there are as many master's scholarship holders as PhD ones. Determine how many scholarship holders there are at each level (bachelor, master's and PhD). 1.25 p
Step-by-step solution
Key idea: in part a) we use the table as the sample space; in part b) we call $g$, $m$ and $d$ the scholarship holders at bachelor, master's and PhD level and translate each sentence into an equation.
a) Simple and conditional probability
Probability of being a master's student and from Science:
Given that the student is from Engineering, the conditioning set is row I (668 students):
b) A system for the scholarships
Let $g$, $m$ and $d$ be the numbers of scholarship holders at bachelor, master's and PhD level. The three conditions:
Substituting the second equation into the first:
Check: $830 + 103 + 103 = 1036$ ✓; the values are also consistent with the table ($830 \le 1333$, $103 \le 533$, $103 \le 134$).