Conditional probability or Compound Probability?

Question

In a village with N people, M are Male and F are female. It was found that X of the male and Y of female are students. If a person is selected randomly, what is the probability that:

A) The person is student and he is a male.

Which formula should I use here?

P(S and M) = X/N

or

P(S/M) = P(S and M)/P(M)

Answer 1

There is no condition here. As it is stated in your question, you are looking for probability that a randomly selected person is both male and a student. Let's look at this from a counting standpoint.

 N - number of villagers
 M - number of male villagers
 F - number of female villagers
 X - number of male villagers who are students
 Y - number of female villagers who are students

From here, we can also extrapolate the following (might be unnecessary, but let's be thorough...)

 X+Y - number of villagers who are students
 N-X-Y - number of villagers who are not students

The key here is the "given" condition. We were already told that $X$ is the number of villagers who are male and students. So the number of students who are both male and students is simply $X$ divided by the sample space, which in this case is all villagers, or cardinality wise is just $N$. Thus

$$P(\text{student and male}) = \frac{X}{N}$$

Conditional probability though puts a condition on the original sample space of $N$ villagers. So by looking at $P(\text{student} | \text{male})$, you are changing the sample space to only male students. Using the conditional probability formula,

$$P(\text{student} | \text{male})=\frac{P(\text{student and male})}{P(\text{male})}=\frac{X/N}{P(\text{male})}=\frac{X/N}{M/N}=\frac{X}{M}$$

But again, think of what the original problem is asking..... out of ALL villagers (not just the male).... this is why the first solution is correct.