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Are they the same rate or are they more common than the main titles?

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The thing about Gold Magikarp is that they actually get rarer the longer you play the game, because as more patterns become available your chance of snagging one gets lower and lower. As you beat leagues you are given better fishing rods, and each rod unlocks a new pattern. Here is the chance of catching a Gold Magikarp with each rod, from Serebii.

Old Rod: 6.67%
So-So Old Rod: 5.74%
Good Old Rod: 5.18%
Great Old Rod: 4.55%
Rare Old Rod: 4.2%
Pro's Old Rod: 4.09%

So, your chances of hooking a Gold 'karp are between 7 and 4 per cent, depending on how much of the game you've completed. If you want to farm Magikarp until you get a gold one, don't beat the Quick League (Mayor Karp rewards you with the So-So Old Rod) to maximize your chance of getting it. Then, farm Magikarp by training relentlessly and throwing them at every berry tree and sunken treasure you can find until eventually there's a Gold Magikarp sitting atop the pile of fish corpses.

Although, if I'm being honest, Gold Magikarp aren't that hard to find. I got one on like, generation eight. Your odds are waaaaay better than in base Pokémon games, so if you keep playing eventually you should get one.

One other note is that the Golden Calico pattern is actually rarer than Gold. Happy fishing!

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