PokéBase - Pokémon Q&A
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Depends on whether you're hunting in the Glimwood Tangle or the Old Cemetary. Either way, authentic Sinistea has a 1% spawn rate out of everything overall, but the rate of forged Sinistea varies, meaning that your ratio of forged/authentic changes.

In Glimwood Tangle, the chance of finding any Sinistea is 11%; 10 for forged, 1 for authentic. Ignoring phases from other Pokemon, this gives you a 1/11 or 9.09% chance of getting an authentic Sinistea.

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The Old Cemetary varies, both for overworld spawns and random encounters. In every weather except Sandstorm, authentic Sinistea has a 1% chance overall, while forged Sinistea has a 19-20% rate via overworld spawns and a 13-19% rate via random encounters. Here are all the rates:

Sinistea cannot be found during a sandstorm.
Clear weather, random encounter: 19:1 ratio; 1/20 or 5% chance
Thunderstorm, random encounter: 13:1 ratio; 1/14 or 7.14% chance
All other weather (except sandstorm), random encounter: 17:1 ratio; 1/18 or 5.55% chance

Blizzard or fog, overworld: 20:1 ratio; 1/21 or 4.76% chance
All other weather (except sandstorm), overworld: 19:1 ratio; 1/20 or 5% chance

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So there’s different authenticity ratios per weather, did you include the in general spawn rate? I’m a little bit confused
For the purpose of spawning Sinistea, the game treats antiques and phonies like separate Pokémon, both with their own spawn rate. The antique Sinistea always spawn at 1% out of all Pokémon in the area, regardless of where you are, while phony Sinistea’s spawn rate varies.
For the purpose of finding the ratio of phony to antique, it makes sense to combine the chances for the two Sinistea forms and then divide the antique form’s 1% by whatever the resulting percentage is.

For example, in Glimwood Tangle, phony Sinistea have a 10% encounter rate overall while antiques are a 1% chance overall. The chance of finding any Sinistea is therefore 11%, with an antique being a 1/11 chance out of that 11%.
Okay, thank you!