How is it determined whether a catch attempt succeeds or fails?

Question

(I have edited this question into a more general form for greater clarity and utility)

This is not a question about how the catch rate is determined, but about how the game determines if a Pokemon is caught after calculating the catch rate. I know about the various formulae that use factors like HP and what type of ball is being used to spit out some catch rate. Let's say it comes out to 10%. When I throw a Poke Ball, how is it determined whether I receive the 10% result, where the Pokemon is successfully caught, or the 90% result, where the Pokemon breaks free?

I assume the game checks an RNG value against the catch rate. I am looking for an answer that describes how the RNG value is generated, how it is checked against the catch rate, and whether the situation can be manipulated for each of the mainline games/generations.

Comments

I have edited this question into a more general form since it might not have been clear as to what I was looking for in an answer. I did eventually catch a static encounter using 1 ball and like 60 soft resets (it was just terrible luck), so the simple answer to whether or not you can do that is "yes", but what I'm really after is learning what's going on underneath the hood that makes it possible and how it might vary by console.

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im pretty sure in at least one gen it runs a rng test for each shake of the ball using an algorithm to check if the shake passes and after 3 passes it catches... not enough for a full answer though

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It's been a while, but I'm almost positive that Pokemon can break out of the ball even after 3 shakes.

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after it catches it shakes three times it catches the Pokemon but a lot  of people said that when they catch the Pokemon and it shakes three times it just breaks out of the pokeball so its quite confusing, but if stars shoot out of the pokeball it means it caught it.

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Cave of Dragonflies have loads of pages explaining the mechanics, maybe something to look into?