Tricky Logic/ Probability Puzzle Involving Venn Diagrams 
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.  
 A: Break the event into disjoint cases:


*

*First student studies History but not Geography; second student studies Geography but not Psychology

*First student studies History and Geography but not Psychology; second student studies Geography but not Psychology

*First student studies all three subjects; second student studies Geography but not Psychology


Compute the probability of each case, and then add them together.
A: As you realized, the probability of the second student studying Geography but not Psychology depends on whether the first student studies Geography or not, and same for Psychology.
So, the easiest thing to do is to consider those 4 possibilities.


*

*The first student studies both Geography and Psychology (in addition to History).


There are $3$ such students (so $P(H_1,G_1,P_1)=\frac{3}{100}$), and with the first student being one of those $3$, there are $37$ students left studying Geography but not Psychology (so $P(G_2,P_2'|H_1,G_1,P_1)=\frac{37}{99}$)


*The first student studies both Geography but not Psychology (but does study History).


There are $25$ such students (so $P(H_1,G_1,P_1')=\frac{25}{100}$), and with the first student being one of those $25$, there are $36$ students left studying Geography but not Psychology (so $P(G_2,P_2'|H_1,G_1,P_1')=\frac{36}{99}$)


*The first student studies Psychology but not Geography (but does study History).


There are $8$ such students (so $P(H_1,G_1',P_1)=\frac{8}{100}$), and with the first student being one of those $3$, there are $37$ students left studying Geography but not Psychology (so $P(G_2,P_2'|H_1,G_1',P_1)=\frac{37}{99}$)


*The first student does not study either Psychology or Geography (but does study History).


There are $7$ such students (so $P(H_1,G_1',P_1')=\frac{7}{100}$), and with the first student being one of those $3$, there are $37$ students left studying Geography but not Psychology (so $P(G_2,P_2'|H_1,G_1',P_1')=\frac{37}{99}$)
So: $P(H_1,G_2,P_2')=$
$P(H_1,G_1,P_1)\cdot P(G_2,P_2'|H_1,G_1,P_1) + P(H_1,G_1,P_1')\cdot P(G_2,P_2'|H_1,G_1,P_1')+P(H_1,G_1',P_1)\cdot P(G_2,P_2'|H_1,G_1',P_1) + P(H_1,G_1',P_1')\cdot P(G_2,P_2'|H_1,G_1',P_1')=$
$\frac{3}{100}\cdot\frac{37}{99} + \frac{25}{100}\cdot\frac{36}{99} + \frac{8}{100}\cdot\frac{37}{99} + \frac{7}{100}\cdot\frac{37}{99}=...$
Now, notice that the only time we really got a change is in the second case: where the first student studies Geography but not Psychology, i.e. where the first student is like the second student in terms of studying Geography/Psychology. And that makes sense: if the first student is not exactly like the second student in that respect, then the number of students that are like the second student, and hence the likelihood of choosing any student like the second student, is unaffected.
Thus, in hindsight, we could have broken this down a little more efficiently in just 2 cases:


*

*The first student studies History, and is like the second student (i.e. also studies Geography, but not Psychology)


There are $25$ such students (so $P(H_1,G_1,P_1')=\frac{25}{100}$), and with the first student being one of those $25$, there are $36$ students left studying Geography but not Psychology (so $P(G_2,P_2'|H_1,G_1,P_1')=\frac{36}{99}$)


*The First student studies History, but is not like the second student (i.e. either does not study Geography, or does study Psychology (or both))


There are $18$ such students ... and in each case, there are $37$ possibilities left for the second student.
So yes, this would have simplified the math ... but the first way was definitely a safer way to go as it did indeed felt like a rather tricky problem.
A: Case $1$: Student $1$ is one of 18 students who dont fullfil the conditions of Student $2$
$\frac{18}{100}\cdot \frac {37}{99}$
Case $2$:Student $1$ is one of those $25$ students not taken in Case $1$
$\frac{25}{100}\cdot \frac{36}{99}$
Add both cases
