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At a certain university, 30% of the students major in zoology. Of the students majoring in zoology, 60% are males. Of all the students at the university, 70% are males. What percentage of the all students at this university are males majoring in zoology?

I tried to get the solution for this question, by calculating:

(0.3 * 0.6) / 0.7 = 0.2571

But the right answer seems to be 18%.

why?

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Let there be $100$ students in the university. Then there will be $30$ majoring in zoology. Of those $30$ students, there will be $0.6*30$ or $18$ males. Therefore the answer is $\frac{18}{100}$ or $0.18$

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To see why this is wrong, take for example, $100$ students in the university. $30$ of them are doing zoology, $18$ of those being male. Can you see why the fact that 70 students are male is unnecessary here? Therefore, you should not be dividing by $0.7$.

What you should do is multiply the probability of being a zoology student with the probability of being a male knowing that the student is a zoology student. So our probability $p$ is given by:

$p = P(M\cap Z)=P(M|Z)\cdot P(Z) = 0.6 \times 0.3$, with:

$M : $ being a male,

$Z$: being a zoology student.

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$$P(A\cap B)=P(A|B)\cdot P(B)$$ In your case, $$A=\text{is male}$$ $$B=\text{is majoring in zoology}$$ so $$P(A\cap B)=(0.6)(0.3)=0.18$$


Notice that $$ P(A|B)\cdot P(B) = P(B|A)\cdot P(A)$$ and so $$ P(B|A) = \frac{P(A|B)\cdot P(B)}{P(A)} = \frac{(0.6)(0.3)}{0.7}=0.2571$$

This is the probability that a student at the university is majoring in zoology given that they are male. This is not the same as the probability that they are majoring in zoology and that they are male.

In the first case, we move all of the males in the university into a big auditorium. We then choose one of them at random, and consider the probability that he is a zoology major.

In the second case (which corresponds to your question), we move all of the male and female students into a big auditorium. We the choose one of them at random, and consider the probability that s/he is a male zoology major.

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