Meta Battle Subway PokeBase - Pokemon Q&A
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I would like to know.

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I'm pretty sure the ratio of shinies to non-shinies is unaffected.

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If a Pokémon with Illuminate is the leading party Pokémon, wild Pokémon are twice as likely to appear.

sumwun is perfectly correct in his concise summary of shiny-non-shiny ratio being unaffected. That is to say, the chance is still 1/8192 for every Pokemon you encounter.

If you're interested in statistics, then you can see how that means you have a greater chance of encountering a Shiny Pokemon due to the sheer increase in the number of trials, as opposed to the probability of the event itself. If you are not much into that, I can interest you in a George Bernard Shaw quote where he says

When I was young, I observed that nine out of ten things I did were failures. So I did ten times more work.

Between you and me, though, if you are interested in getting a shiny Pokemon, Illuminate does help out; just not in a direct manner. The chance is still the same.

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Is there not ANY way to get Shiny Minior easier?!
Shiny charm, Masuda method
By the way, the chance in Generations 6 onward is 1/4096.