How do I know if two events are independent or not? Let's say we flip a coin twice:
The sample space (set of all possible outcomes) is:
$$\Omega = \{HH, HT, TH, TT\}$$
Now let's say we have the following two events (sets of outcomes):
$$A= \{HH, HT, TH\}, B = \{HT, TH, TT\}$$
where $A$ is the event that at least one coin is heads and $B$ is the event that at least one coin is tails.
Now the intersection between A and B ($A\cap B$) is the set of outcomes that are shared between A and B,  the event (C) where at least one coin is heads AND at least one coin is tails:
$$C = A\cap B = AB = \{HT, TH\}$$
In this trivial example, I can calculate the probability of C by taking the size of the event over the size of the sample space: $$P(C) = \frac{size(C)}{size(\Omega)} = \frac{2}{4} = 0.5$$
But I am struggling to understand how I would find this answer if the sets were too large or complicated to count out.
The calculation $P(A)*P(B) = \frac{9}{16}$ clearly gives a different answer. I think this means that events A and B are not independent. Two events are dependent if the outcome of the first event affects the outcome of the second event, so that the probability is changed. So conceptually, how/why is A affecting B (or vice-versa)?
And more generally, if it's not conceptually clear that one event is influencing another, does that mean the only way to determine if they are independent is by collecting data from observations?

I have a potential answer to my question.
I think the issue I am having is conceptualizing the definition of "dependent events."
I think this is a more useful definition: Events are dependent if the outcome of one affects the probability that the conditions of the other event will be satisfied (AKA, that it would occur).
In the example in the question, we can see that the outcomes of event A will indeed affect the probability that event B will occur. If the outcomes of A are either HT or TH, then the probability of B occurring is 100%. That is because the conditions of B, at least one Tail, is satisfied by both those outcomes. If the outcome of A is HH, then the probability of B occuring is 0%. Again, because the conditions of B, at least one Tail, is not met by that outcome.
