[Math] Tricky Logic/ Probability Puzzle Involving Venn Diagrams

Question

A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and that the second studies Geography but not Psychology.

I am lost when it comes to this problem, as I thought you would need to add up the number of students who study history overall, which is 43. Then I would simply multiply 43/100 by the number of students who take Geography but not Psychology, however this leaves an uncertainty as to whether the first student was from the 25 who take H and Geography or not, and this would change the answer I get.

If you could explain what I need to do and why in simple terms, I would be grateful.

If there are any issues with the question, please let me know.

Answer 1

Break the event into disjoint cases:

1. First student studies History but not Geography; second student studies Geography but not Psychology
2. First student studies History and Geography but not Psychology; second student studies Geography but not Psychology
3. First student studies all three subjects; second student studies Geography but not Psychology

Compute the probability of each case, and then add them together.