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I recently downloaded Pokemon Shuffle, and not too long ago, I tried catching a Pokemon on the Expert stage. It had at first a 20-ish % catch rate (failed), and then with the great ball, had a 68% chance (also failed). After that, I tried catching the Special Limited Event Pokemon. It only had a 5% catch rate, but I caught it first try. There is only about a 3 percent chance of this happening. Is there some special mechanic to the catch rates, or is pure luck?

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So I'm not the only one to experience this? I succeeded with 5%, after failing with 90%
I used to have this happen to me occasionally, but otherwise I thought the percentages were valid indicators.
I thought for sure the game just rolled a number between 1 and 100.
You know those bars that appear in some games? The ones with an arrow moving back and forth along the bar and you have to click at the right time to make the arrow stop in the green section of the bar but not the red section. I think the Shuffle catch rates work like that. Imagine that there is an invisible arrow moving across the catch rate bar and lets say you have a 25% catch rate, that is the green section, and 75% of the bar is the red section. When you tap the Pokeball you stop the invisible arrow on the somewhere on the bar. That's why you can fail with 90% because the arrow lands on the 10% you don't have, and that's why you can catch with 5% because the arrow lands on the 5% you do have. It all comes down to luck, that's what I think anyway.

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Extra catch rate percentage is different depending on how many moves you have left (Or time, in expert mode) when you defeat the Pokemon. For example, Absol have a 1% catch rate. For every 3 or 4 seconds remaining when you defeat it, it has a 5% better chance of catching it.
Here is a website I found very helpful for Pokemon shuffle. (Translate it first)
https://www.pkparaiso.com/shuffle/lista-de-pokemon.php
The answer is, that the percentage is just giving you an "x out of 100" chance.
Whether or not you catch It is completely up to the game. Once I caught a Wobbuffet with a 2% chance, but on the other hand, I couldn't catch a 98% chance Nidorina.
It's just the luck.
Hope I helped! :)

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