Should 2% be considered intersection or conditional probability?

Question

My question is based on Sheldon M Ross' 'First Course in Probability'

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that

1. the student is female given that the student is majoring in computer science;
2. this student is majoring in computer science given that the student is female.

My Approach:

Let F = Random selected student is a female, so $P(A) = 0.52$

C = Random selected student is majoring in Computer Science, so $P(C) = 0.05$

and Probability that 2% women are majoring in computer science is equivalent to probaility of random selected student is doing computer science given she's a woman. Therefore, $P(F|C) = 0.02$

End

I am aware that the above conditional approach is wrong. But why is it so? According to a solution I found over the internet, it should be $P(F \cap C) = 0.02$.

But why? I want to understand the intution behind this and approach questions more of this type.

Thanks!

Answer 1

You are told in the question

Two percent of the students are women majoring in computer science.

You are misinterpreting this statement to be a follow-up to the previous sentence; i.e., you are thinking that it means

Among students majoring in computer science, two percent of the students are women.

But the sentences are separate because if the author had meant the second (conditional) interpretation, he would have written it that way.

So $\Pr[F \cap C] = 0.02$.

Furthermore, the first part of the question would be trivially answered under your interpretation.