When to use semicolon and comma in a list of clauses?

Yu-Cheng Chen!

We usually use commas in lists within a sentence:

I believe that dogs are fun, cats are lazy, and pets are a lot of work.

Semicolons are used between to highly connected thoughts, almost like a conjunction:

He ran all night and was utterly exhausted by the ordeal; he collapsed.

So, the better formulation given your example is the first:

The probability that event A equals to 1, B equals to 2, and C equals to 3 is X.

That's because our brains conceptualize this as:

The probability that event 'A equals to 1', 'B equals to 2', and 'C equals to 3' is X.
The probability that event 'A=1', 'B=2', and 'C=3' is X.

As a math guy, I'd probably write:

The probability that event A=1, B=2, and C=3 is P.

Note well, in very sophisticated writing, sometimes semicolons are used, but rarely in common usage.

Be such that the claims in the book need to be addressed, which is the purpose of this exegesis, then one can conclude certain ideas are tenable such as: the nature of man must be explored soley by philosophers who by their method are more apt to the characteristics of man clearly, no insult meant to other professions; that to explore any collection of human traits is a process which requires time, energy, and a peculiar attention to the psychological is beyond doubt; and it is no small victory for a personality framework to include such a comprehensive and sweeping set of ontological and epistemological features, and thus I give credit where credit is due.

Semicolons go between two parts of a sentence when each is technically a sentence. "He was victorious in yesterday's mayoral race; work begins now."

Among stops (I am using the term here to denote a class of punctuation marks, not consonants), the semicolon is intermediate between the comma (“,”) on the one hand, and the period (“.”—full stop in British usage) on the other. For usage in separating independent clauses, think of it as a lesser period, giving the reader permission to take a slightly lesser mental/grammatical processing break. For usage in lists, think of it as a greater comma, separating list items that cannot be separated merely by commas because at least one of them includes a comma, to separate the main naming of the item from some appositive, non-restrictive relative clause, or other parenthetical addendum. The point I wish to emphasize here is that your confusion is a confusion between these two altogether different usages of the semicolon. This page from Merriam-Webster may be helpful.