I'm learning about conditional probabilities and the Bayes theorem, but I don't know how to really think about it. The problem I have is the following:
There are 50 men and 40 women at a workplace. 10% of the men and 2.5% of the women smoke. When choosing an employee at random, we happen to choose a smoker. What is the probability that the smoker is female?
My solution idea is the following, but I'm not sure it's correct.
Let M = men, F = female, S = smoker.
We want
P(F | S) = P(F $\cap$ S) / P(S)
Here
P(F $\cap$ S) = 0.025
P(S) = (0.1*50 + 0.025*40) / 90 $\approx$ 0.0667.
Thus:
P(F | S) $\approx$ 0.025 / 0.0667 $\approx$ 0.37
What makes me suspicious of that this might be wrong is that I'm uncertain if it's correct to say P(F $\cap$ S) = 0.025. It may be that this should be P(S | F) meaning that I could use the Bayes theorem P(F | S) = P(S | F) P(F) / P(S).
Any hints are much appreciated!
Construct a tree illustrating the problem.
$$\begin{cases} men \ \ 5/9 & \begin{cases}10\% \ smoker \\ non-smoker \end{cases} \\ female \ \ 4/9 & \begin{cases} 2.5\%\ smoker \\ non-smoker \end{cases} \end{cases} $$
If you compute probability of intersections. For example $P(F\cap S) = 4/9\cdot 0.025$ because when considering the event $F\cap S$ we must keep in mind the probability of $F$ and that of $S$.
On the other hand if we have $P(S|F)$ then we have the fact that the person is female as an assumption (just like it would have probability $1$) so we do not look at the $4/9$ probability hypothesis.
In conclusion, if you use Bayes you get $P(F|S) = P(F\cap S)/P(S)$. We have $P(F\cap S) = 4/9\cdot 0.025$ and $P(S) = 5/9\cdot 0.1+4/9\cdot 0.025$.
Therefore $P(F\cap S)$ is not $0.025$. That is the value of $P(S|F)$.