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Does Quick Ball gain effectiveness against a Pokemon if the Pokemon your trying to catch is damaged? This might be a weird question. The only way in any Pokemon game to throw a Quick Ball first turn at a damaged Pokemon is against a roaming Pokemon. So does a Quick Ball gain effectiveness in this otherwise impossible scenario and if so, what are the odds of catching a legendary at 1 HP with a first turn Quick Ball

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Please look at the catch rates of legendaries. The only Legendary Pokemon that don't have a catch rate of 3 are story legendaries
Nah they were asking about if you could throw a Quick ball at the start of every encounter and it still had the effectiveness. I know that is true, I'm talking about if quick ball gains effectiveness with damage

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I'm going to be using Gen 8 Mechanics. I'm going to be using raikou, both Pokemon will be at level 100, and there will be NO status.

You have a 5.24% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 13 balls and at least a 95% chance of catching it within 56 balls.

So, this is what I got when you yeet it when it has no damage. Now, what if it has, say, stealth rock chip? (I know it won't, that's just an example)

You have a 6.157% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 11 balls and at least a 95% chance of catching it within 48 balls.

Now that does change it. What about something else? Say, 50% chip.

You have a 8.806% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 8 balls and at least a 95% chance of catching it within 33 balls.

What about 1 HP?

You have a 11.925% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 6 balls and at least a 95% chance of catching it within 24 balls.

Thus, it does seem to change. Hope this helps :)

Source

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Okay, but wouldn't that still make the calculations wrong? In my opinion, accuracy is more important than relatability, plus, a bunch of people don't have switches yet, so it wouldn't be relatable to them.
The person is not giving a specific pokemon they want to be caught, first of all, so they presumably do not care which generation mechanics I use along with which pokemon. If they have any sort of problem with it, I am happy to change it. Plus, I personally started with the gen 8 mechanics, and I'm sure a lot of people have, and a lot of people are getting in to the math and the formulates of it more in gen 8, as shown by the increase of math related questions on the site.
Okay, that's fine, nevermind.
Ok, so first off being Gen 8 mechanics or Gen 4 mechanics doesn't matter, because the Quick Ball has not changed at all since it's release.

If you haven't got a switch yet its your fault it's been out for four years...

Every roaming legendary has the same catch rate, (Raikou, Entei, Suicune, Articuno, Zapdos, Moltres, Mesprit, Cresselia, Latias, Latios have a catch rate of 3.)
According to serebii, it has, before gen 7, the rate was 4, after gen 7, it was 5. Also, I knew about the catch rate thing, I was talking about the ball factor or whatever it's called. Sorry, let's just stop arguing.
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Yes, it is possible to have the hightened chance of quick balls, if they happen to not do the thing that roaming legends sometimes do where they run away before everything. As for that statistic, assuming nothing else is wrong with it(no status condition or anything.), the chance would be dependent on it's max HP, since that's how the catch rate formula works. Therefore it'd be a different chance depending on what legendary. However, let's just say that the Pokemon's HP is about 100.(I just calculated the Base Stat of all Roaming Legendaries after Quick Balls where made in Gen 4), the procedure would be:

Also, since all roaming legendaries are pre-Sun Moon, I'll use 4 as the ball factor.

3 x 100=300-2=298 x 3=894 x 4= 3872/300=12.90......

So, on average, the chance of catching a roaming legendary in a quick ball at 1 HP would be: 12.9%

Please correct me if I'm wrong in my Math.

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Two Notes, when doing my average calculations, I rounded up, it was actually something like 99.77 something, and secondly, I feel like I did the first part with the HP Current wrong, because that's just 1 x 2, which seems a bit off.
Also, it's to be noted, that most of these legends will be easier than that, most are at 90 or below, with the exception of Cresselia in Sinnoh.