History and Geography are two of the subjects students may decide to study - HSC - SSCE Mathematics Advanced - Question 14 - 2020 - Paper 1

Using the information from part (a):

We have:

• Total students studying Geography: 18
• Students studying both History and Geography: 5

Total students studying only Geography:

$G_{only} = 18 - 5 = 13$

The probability that a student studies Geography does not study History is:

$P(\text{not } H | G) = \frac{G_{only}}{G} = \frac{13}{18}$

For the first student, we know:

• Number of students studying History: 20
• Total students: 40

The probability that the first student studies History is:

$P(H) = \frac{20}{40} = \frac{1}{2}$

For the second student:

• After the first student has been chosen, there are now 39 students left.
• If the first student studied History, then 19 students remain studying History and 20 students do not study History.

So, the probability that the second student does not study History is:

$P(\text{not } H) = \frac{20}{39}$

Thus, the combined probability of both events occurring is:

$P(H) \times P(\text{not } H) = \frac{1}{2} \times \frac{20}{39} = \frac{10}{39}$