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

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!

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.

• Actually there is enough information to decide they are not independent as the number of men is $11$ and the number of women $14$, which are coprime so the age distributions cannot be the same. But, as you say, this does not matter. – Henry Feb 12 '20 at 13:50
• @Henry Oh, good point. – lulu Feb 12 '20 at 13:52