I would like to make a disclaimer that I am still relatively new to probability, if I have used some incorrect naming convention(s) for probability, I do apologize.
The Problem
Suppose you have b number of non-mutually exclusive events such that $P(A_{1}) = P(A_{2}) = ... = P(A_{b})\text.$
Is there a "simplification" (perhaps this is the wrong word) for
$$P(A_{1}\;or\;A_{2}\;or\;...\;or\;A_{b})?$$
Previous Research
I looked online and was unable to find anything helpful or that I could understand, except for the fact that for non-mutually exclusive events A and B,
$$P(A\;or\;B)=P(A)\;+\;P(B)\;-P(A\;and\;B).$$ $$P(A\;and\;B)=P(A)\;*\;P(B|A)$$
My Personal Attempt
I gave many goes at simplifying this, and this was my best attempt yet:
EDIT: As Henry has pointed out in the comments for this post, I was mistaken in some of my assumptions. Therefore, my attempt was wrong. I have also changed the Previous Research section to state correct statements.