Can a number ever be small enough to be considered zero?

[Siegmund]

That depends on who makes the first move. If the casino announces fixed odds, you can choose a number of trials great enough that you are likely to win. If, on the other hand, you announce how many times you are willing to play, they can create a game with bad enough odds you are unlikely to win.

But there is no "one size fits all" number which can be names as "close enough to be considered zero." It's a case by case analysis to determine how small of a number, if any at all, is negligible for a particular type of mathematical question.

In mathematical proofs you will often see phrases like "for sufficiently large N," which is a shorthand for a particular condition -- usually something like "if you first specify the maximum size of error you will tolerate and call it delta, I can name an N beyond which you're guaranteed to always come closer than delta to the answer."