How do you generally tell the independence of events in a probability problem when it is not outright stated?

Mathematics Asked on January 7, 2022

First question, do probability problems generally consisting of two categories always assume two events are dependent?

Let’s say there are two categories from a set of n people. They are gender, divided into $M$ – male and $F$ – female and course which are $E$ – engineer and $D$ – doctor.

I will assign variables for each of the four unions.
$$M ∩ E = a$$
$$F ∩ E = b$$
$$M ∩ D = c$$
$$M ∩ D = d$$

My question is, why is that with these kinds of problems we always go for conditional probability? Is it because they have intersection as given? From my conceptual understanding of dependent events, I don’t think gender and preferred course are suppose to intuitively be related.

Okay someone might argue they could be. But why is it if like the problem involves x category and y category and they just give out values we just always go for the idea that they are dependent?

The bigger question is, how do you if a problem uses dependent or independent events without just relying on common types of problems?

Is it right like that if two things can happen at the same time for like a person or product, we just go for it being dependent? But if two things can happen at the same time but they do not pertain to the same subject, they are independent? Like the event that it rains in City A and it is sunny in City B is independent because they can totally happen at the same time but they don’t really talk about the same thing?

The independence of events depending on their subject was just an assumption of mine. Is this true or not? Is there any generalization on how to tell the independence of two events in the problem when it is not outright stated like in the one I mentioned above?

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