It's 1/4,096, the same way it's been since XY.
A few notes:
-You can't soft reset for shiny Legendaries or shiny starters in SW/SH. You can, however, Masuda Method for shiny starters.
-High leveled Pokemon in the wild-area are also shiny locked as to prevent trainers from encountering shiny Pokemon they are most likely unable to capture.
-Pokemon in the overworld don't appear as shiny until you encounter them in battle.
-Like in LGPE, you can increase your rates of finding shiny Pokemon in the wild by capturing the same Pokemon over and over.
-You can also slightly increase your odds of encountering a shiny by battling the same Pokemon often.
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