One of the philosophical ideas of maths is that there are multiple ways of looking at a problem, and those multiple ways should feed back onto each other to confirm your answer.
Maths is not just a bunch of rules that have to be followed without question: in this case, maths is modelling something practical and the answer needs to match reality. Without that, the probability model could not even have been invented. If you can't get beyond the stage of do_question+check_answer, you are never going to be capable of anything more than grinding out other people's mathematical results.
I'm sorry you weren't taught maths by this person.
I notice this horrible effect every time I'm solving exercises for uni and there are no solutions. I think I've done good work. Sometimes, everything really was right, other times, I made fundamental mistakes all across the exercises, showing me that I didn't understand something important.
Of course, when you're in research, there's no key. That's why it's so difficult. But it doesn't make sense to reinvent the wheel, as in, researching things that are already well known.