GRE Probability - How to tell if Bayes Theorem or Independent Events?

Question

I have a question about the below GRE Quant sample problem:

One person is to be selected at random from a group of 25 people. The probability that the selected person will be a male is 0.44, and the probability that the selected person will be a male who was born before 1960 is 0.28.

Quantity A: The number of males in the group who were born in 1960 or later
Quantity B: 4

A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined from the information given

I have solved correctly using the following method:

P(Male, Born before 1960) = 0.28
P(Male) = 0.44

P(Male) - P(Male, Born before 1960) = 0.44 - 0.28 = 0.16

0.16 = $\frac{x}{25}$ => x = 4 => C is correct.

My question is that I initially paused on this question because I wasn't sure if I was correct in my assumption that being male and being born before 1960 were independent or dependent conditions- i.e. I wasn't sure if I should be using Bayes Theorem or the multiplicative law of independent events. I wound up being correct that they were independent events, but am concerned that I don't know why.

Am I overthinking this? It's been awhile since I learned this material so please forgive me if I'm making a silly error!

Answer 1

Your argument does not use independence.

All you use is the Law of Total Probability which, here, just tells us that $$P(E_1)=P(E_1\cap E_2)+P(E_1\cap E_2^c)$$

Here $E_1, E_2$ could be any two events but in your case $E_1$ is "being male" and $E_2$ is "being born before $1960$".

As remarked by @Henry in the comments we can in fact decide the dependence:

Suppose the events were independent (we will derive a contradiction)

We know there are $.28\times 25 = 7$ men born before $1960$ and $4$ born after, so $11$ men all told. Let $N$ be the number of people (of either gender) born before $1960$. Assuming independence, we should have $$P(E_1\,|\,E_2)=P(E_1)\implies \frac 7N=\frac {11}{25}\implies N=\frac {7\times 25}{11}\notin \mathbb Z$$

So we see that the events can not be independent. As remarked, however, this is irrelevant to the given problem.