Find the probability using Bayes' Theorem

Question

You, a woman with a medical background, are one of 198 applicants for an MBA programme of whom 81 will be selected. You hear, along the grapevine, on good authority that there were 70 women applicants, of whom 38 were selected. Assess your probabilities of being accepted before and after you receive the grapevine information.

What I've tried:

For the first part, it's straight-forward that the probability of being selected before hearing the news is $\frac{81}{198}$.

Now, the second part is where I'm stuck. From what I understand, we need to find the probability of being selected given that it's a woman.

So, let's say that $A$ is the event of being a woman and $B$ is the event of being selected.

Therefore, we want to find $$P(B|A) = \frac{P(B \cap A)}{P(A)} = \frac{\frac{38}{70}}{\frac{70}{198}} = 1.53$$

As you can see, this is clearly wrong. I'm not sure what I'm doing wrong. an someone please help me.

Thanks

Answer 1

When you compute $P(B \cap A)$ -- the probability that a given applicant is a selected woman -- you should have your denominator be over all applicants, not only women. (Just as when you computed $P(A)$, the probability that a given applicant is a woman, you had your denominator over all applicants.) So it should be $\frac{38}{198}$, not $\frac{38}{70}$.

Answer 2

You have the Information that you are a woman and that 38 of 70 women wer accepted. SO the probability, given that the Information is correct, is 38/70.