For a probability of the form P(A) x P(B|A), does the P(B|A) part express the probability of event B given ONLY event A? Does this probability completely ignore the other factors and events, especially in a complex scenario, and treat them like they don’t exist?
Simply using the multiplication rule of probability in a realistic scenario doesn’t seem to make sense. For example, let’s say I want to find the probability of me both passing the exam and being happy. From what I learned, this would be expressed as :
P(Passing exam) x P(Being happy | Passing exam)
It seems reasonable that the second part of this expression is a high probability and, let’s just say the probability of the first part is also high :) This should, supposedly, give me the probability of both passing the exam and being happy. But here’s the thing: there are so many other factors that we need to consider! I can’t just ignore these, right? So my question is really just about whether or not this supposed probability assumes no other given factors besides “passing exam” or, to put it generally, the given factor.
Yes, $\mathbb P(B\vert A)$ is the probability of $B$ in an alternate universe where you have discarded any outcome that does not belong to $A$.
I am not sure to understand your concern. What do you mean by there are many other factors that we need to consider? Do you mean that it's not that easy to pass the exam? In that case your P(passing exam) would not be that high. Or do you mean that it is not that trivial to be happy just because you passed the exam? In that case your P(Being happy | Passing exam) would not be as high as you say it is.
Maybe you get confused between P(Being happy | Passing exam) and P(Being happy of passing exam).