# Conditional probability or Compound Probability?

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)

• It doesn't say that "what is the probability that the random person chosen is a student, given that his is a male..." That would be conditional. That would be the second formula. Your first formula is correct. Dec 11, 2019 at 2:35
• @Eleven-Eleven I too think that my first formula is correct but my teacher told me that the second one is correct. So, I want to confirm it here. He said that 'and he is a male' says the selected one is already male. Dec 11, 2019 at 2:41
• The probability is indeed $X/N$, as the question is written. The probability in question is "the person is student and he is a male," not "the person is a student, given that he is a male." So your professor is mistaken and perhaps worded the question incorrectly. Dec 11, 2019 at 2:55
• @Math1000 Thank you both ! Now I am confident that I am correct. Dec 11, 2019 at 3:37

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.