These are several questions, I would like answers to all of them.
What is the algorithm of meeting a Pokemon in the wild?
What is the algorithm of meeting Pokemon in the wild with a specific nature?
What is the algorithm of meeting Pokemon in the wild with a specific nature while having a Pokemon with Synchronize?
What are the chances of meeting a shiny Pokemon with a specific ability (with two to choose from), a specific nature, 6 flawless IV:s and having a specific gender?
Sorry for the trouble but I would like to know.
Thank you in advance!
The formula to meet a Pokemon in the wild is simply defined by P = x / 187.5. x is either 10 (Very common Pokemon), 8.5 (Common Pokemon), 6.75 (Semi-rare Pokemon), 3.33 (Rare Pokemon), or 1.25 (Very rare Pokemon). That gives either a 5.33%, 4.53%, 3.60%, 1.78%, or 0.67%, respectively. (Source)
There are a total of 25 Natures, meaning the chance of encountering a Pokemon with a specific nature is 1/25, or 4%. So, multiplying the encounter rate by that factor will result in the equation P = x /4687.5. This will turn then abformentioned encounter rates into 0.21%, 0.18%, 0.14%, 0.07%, or 0.03%.
Synchronize turns that 1/25 chance into a nifty 1/2 chance, which will greatly enhance the encounter rate. The formula now becomes P = x/375. The encounter rates now turn into 2.67%, 2.27%, 1.80%, 0.89%, or 0.33%.
At this point, I have no idea why anyone would bother, but...