PokéBase - Pokémon Q&A
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Say for example you find 78 Kabuto without shiny in Pokémon GO, does the odds chance to 78/4096? Or what about if you get 689 encounters without a shiny in UM? Does the chance go up to 689/4096? Is this the case, or is shiny chance just random with the odds?

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The shiny chance is just random with odds. If it increased as you are asking, then you would be guaranteed a shiny by the 4096th encounter, (4096 encounters would make the chance go up to 4096/4096, which is 100%) and this is definitely not the case.

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The chances are always at 4096, its mainly luck , the more envounters you get USUALLY you will gte the shiny , bit the actual odds are always the same.

Source:knowledge and experience

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It's 1/4096, not 1/4000.
I just rounded it as did Mega-Roserade. Not much of a difference
In fairness to you both, the asker was the one had it wrong to begin with. I fixed every post.