Calculating the conditional probability Prob(A|B)

Question

There are 6 women philosophy graduate students in the philosophy department, 12 women philosophers, and 20 philosophy graduate students. What is the probability that a philosopher is a graduate student, given that she is a woman? What is the probability that a philosopher is a woman, given than (s)he is a graduate student?

I am having trouble understanding how to calculate this.

For the first question, do I divide 6 (the event of being a philosophy graduate student) into 18 (the event of being a woman)? Or do I divide 20 (the event of being a graduate student) into 18 (the event of being a woman)?

For the second question, do I divide 18 (the event of being a woman) into 20 (the event of being a graduate)? Why? Or am I completely wrong and need to calculate the probability of being a woman philosopher (18/38) and the probability of being a grad student (20/38) separately?

How would I solve this problem???

Answer 1

You can use the definition $P(A|B)=\dfrac{P(A\cap B)}{P(B)}$. In a finite situation, this becomes $P(A|B)=\dfrac{N(A\cap B)}{N(B)}$.

Let us set $A=$ Woman and $B=$ Graduate Student.

Then your first question is $P(B|A)=\dfrac{6}{12}=\dfrac{1}{2}$.

You should be able to figure out your second question using the same reasoning.