Question

6. a. Sixty students in a class took an examination in Physics and Mathematics. If 17 of them passed Physics only, 25 passed in both Physics and Mathematics and 9 of them failed in both subjects, find i. the number of students who passed in Physics ii. the probability of selecting a student who passed in Mathematics 17​

Answer 1

Let $C$ be the set of all students in the classroom.

Let $P$ and $M$ be the sets of students that pass physics and math, respectively.

We're given

$n(C) = 60$

$n(P \cap M') = 17$

$n(P \cap M) = 25$

$n((P \cup M)') = n(P' \cap M') = 9$

i. We can split up $P$ into subsets of students that pass both physics and math $(P\cap M)$ and those that pass only physics $(P\cap M')$. These sets are disjoint, so

$n(P) = n(P\cap M) + n(P\cap M') = 25 + 17 = \boxed{42}$

ii. 9 students fails both subjects, so we find

$n(C) = n(P\cup M) + n(P\cup M)' \implies n(P\cup M) = 60 - 9 = 51$

By the inclusion/exclusion principle,

$n(P\cup M) = n(P) + n(M) - n(P\cap M)$

Using the result from part (i), we have

$n(M) = 51 - 42 + 25 = 34$

and so the probability of selecting a student from this set is

$\mathrm{Pr}(M) = \dfrac{34}{60} = \boxed{\dfrac{17}{30}}$