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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???

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    $\begingroup$ Welcome to Math.SE! When asking a question, please include a look at what you have tried / where you are getting stuck. You'll find you get much more help that way, because we know exactly what idea is giving you trouble! $\endgroup$ Jul 17, 2013 at 18:40
  • $\begingroup$ 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??? $\endgroup$
    – sunny
    Jul 17, 2013 at 18:53
  • $\begingroup$ I've added your comment above to your question. Note: it is possible for you to edit your question. $\endgroup$ Jul 17, 2013 at 19:22
  • $\begingroup$ It is clear that they count philosophy grad students as philosophers. In England, even undergraduate math majors are called mathematicians. $\endgroup$ Jul 17, 2013 at 19:22

1 Answer 1

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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.

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