How can I solve this probability problem? A most unusual Irish pub serves only Guinness and Harp. The owner of this pub observes that 85% of his male costumers drink Guinness as opposed to 35% of his female costumers. On any given evening, this pub owner notes that there are three times as many males as females. What is the probability that the person sitting beside the fireplace drinking Guinness is female?
So what I did was
Males that drink Guinness are 0.85
Females that drink Guinness are 0.35
There are also 0.75 males and 0.25 females
So I took everyone that wasn't female and drinks Guinness by doing this:
0.75 + (0.25 * (1-0.35) = 0.9125
So I subtract this to 1
1 - 0.9125 = 0.0875
And so I think the answer is 0.0875 or 8.75%.
Is this correct or am I doing something wrong?
 A: What is wrong with your attempt has been elaborated in the comments : you calculated only the probability that the person is Female and drinking Guinness. You have not taken into account the fact that it was known that the person was drinking Guinness. This will increase the probability significantly.
See, the person you see is drinking Guinness. Therefore, the desired probability is the probability that the person is female GIVEN that he/she is drinking Guinness.
Thus, if $F$ denotes the event "the person is female" and $G$ denotes the event "the person is drinking Guinness", then we need $P(F | G) = \frac{P(F \cap G)}{P(G)}$.
To calculate $P(F \cap G)$, we need to interpret what we have. It is known that $85$% of the male customers drink Guinness, so $P(G|M) = 0.85$. Similarly, $P(G | F) = 0.35$. Additionally, $P(M) = 0.75$ and $P(F) = 0.25$.
From here, $P(F \cap G) = P(G | F)P(F) = 0.0875$ (as per your calculation).
On the other hand, $P(G)$ is what we don't know. We do it using the Bayes' rule : condition on whether the person is a man or woman, and multiply by the respective probability of the person being man or woman. That is :
$$
P(G) = P(G | F)P(F) + P(G |M)P(M) = 0.0875 + 0.85 \times 0.75 = 0.725
$$
Therefore, the actual answer is $\frac{0.0875}{0.725} = \frac{7}{58}$.
