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Are the encounter rates found by data mining the games, by trial and error with data, or from the physical guides they made for the games? Just a general curiosity after looking at different sites for encounter rates and noticing one was off (due to a typo most likely).

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IDK, maybe calculated?
I think official guide books have encounter tables in them.
Official guide books have rates like very common, common, rare, and very rare, but as for like the exact percentages no
I'd assume datamining.

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Most likely datamining, because it would take an unbearably long time to calculate those from trial and error like you mentioned. Datamining would give the exact answer and would be way simpler than calculating.

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How do you know there are no easier ways?
Well I am assuming that datamining is what is done but, if anyone finds anything else to the contrary let me know.