So I took a machine learning module during my final year at university and I struggled quite a bit with "ambiguous" mathematical notation. E.g. the slides used by our lecturer occasionally had typos in them. Sometimes he'd use an italic X and sometimes he'd use a normal X to refer to the same thing. I don't think anyone in the class cared about such details but I spent more time worrying about all of this instead of actually learning.

I recently decided to give machine learning another attempt and I'm again worrying about trivial issues. In one of the books I've been reading, the author introduced a variable y (lowercase) but then referred to it at the start of a sentence and used an uppercase Y (probably not his fault). It's a really trivial thing that I'm guessing most people won't care about but it really started bothering me because I kept asking why an uppercase Y has been used. Is this a typo or am I misunderstanding something? It made me doubt my ability to understand maths.

I recently started reading this book:

http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/020217.pdf (legal copy made available by the author)

On page 8 (or absolute page 32), the author says that "we will use the shorthand p(x, y) for p(x and y)" and I think great... p(x, y) is a probability. On the same page, he then says "given a joint distribution p(x, y)"... But aren't probability and distribution two separate things? Is this usage of notation something I have to come to terms with or am I missing something and the author is completely correct?

I thought it was just me who worried about all of this until I recently came across this quote on math.stackexchange:

"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna

I'm not sure if this is referring to the same problem I'm having or something completely different but it made me wonder perhaps I just need to learn to overlook this and I'll get used to it in time. I tend to understand concepts fairly quickly once notation starts making sense or I ask someone to explain it to me in plain English. I'm just spending too much time worrying about making sure I accurately understand everything.

So my question is whether or not this amount of ambiguity normal or is there an error in my understanding of maths that is likely to confuse me. When I encounter an issue like this, should I keep on reading and tell me brain "it's all going to be okay, keep reading" or spend ages making sure I've understood everything? I've been told by quite a lot of people that I tend to overthink and worry too much so I wanted to ask for advice on here. Thank you!