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This may sound crazy, but I feel like catching isn't based on luck alone. Sometimes, I can't catch it with an ultra ball, but when I use a normal ball it instantly catches. Also, I feel like I catch more when I mash the "A" button. It may just be me on that last one, but I do think that the first one actually works, based on the amount of times it has worked. Do either work? Sorry if this breaks rule 1.4, that they'd need access to the code and stuff to answer it.

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After modifiers such as the ball used, level of the Pokémon, HP, and status, it’s is RNG.

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Catching is not based on luck alone, but by a variety of factors, the most prevalent of which is RNG, some of which change generation to generation. Some of these factors include:

  • HP
  • Level
  • Status
  • All species have a specific catch rate. For example, most legendaries are 3.
  • The type of Ball
  • Pass Powers, O-Powers, or Rotom Powers
  • Being in an initial capture spot, such as Route 101 in ORAS.

Many more factors would be included in this list, but most are generation(s) specific. These factors are plugged into a mathematical equation that calculates probability (aka RNG). You can learn about these equations and factors more on this Catch Rate Bulbapedia page. Here's an example of the Gen VI catch rate equation:

Hope I helped!

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What about normal balls sometimes working better than ultra balls?
They don't
that could be explained by selective memory
"normal balls sometimes working better than ultra balls" can easily be explained by rolling a six sided die vs a 100 sided die. If the number you want is 6 and you roll the six sided die three times and don't roll a 6, but roll the 100 sided die once and immediately get 6, that's not the 100 sided die being more likely to roll a 6, that's literally just how statistics works. Just because you happen to occasionally have better luck with something that's statistically worse doesn't make it better.
IIRC, The great balls were worse in gen 1 due to a programming error, but that’s the only instance of worse balls being better than the good balls.