Question

What are the chances of running into a shiny Pokemon with its hidden ability?

Comments

Not a full answer, but https://pokemondb.net/pokebase/278890/how-do-i-get-a-pokemon-with-its-hidden-ability

Answer 1

This is currently unknown due to the chance of getting a Hidden Ability Pokemon being unknown. The chance of getting a shiny Pokemon is 1/4096 (3/4096 with a Shiny Charm) and 5/4096 in horde battles (7/4096 with Shiny Charm).

Source
Source (Note how it says "small chance")

Answer 2

First off the only way to find a Pokemon with its hidden ability(besides hordes) is the friend safari, which you can get Pokemon with hidden abilities. Plus the Pokemon can also be shiny. So... the chance of finding a shiny was halved, so it's 4105(estimate) and a 1/3 or 1/2 chance of finding a Pokemon with its hidden ability(depends on the Pokemon, again, just estimating). So... after a lot of math in my head, I'm guessing the chance is about 3.02% to 6.04%.
Hope I helped!