[talk] The highest probability to find a shiny in a hunt is on the first encounter.
Let's say I am going to start a hunt with a 1/512 probability to catch a shiny. But just before the beginning of the hunt, I give to everyone 1$ to bet on which number of encounters you think I will find the shiny. Not "at least n encounters" but "exactly n encounters". Everyone has to bet on the exact number they think I will find the shiny, and when I start the hunt, I continue until I find the shiny and then stop the hunt.
Betting on the first encounter is the best bet in this game, and I tell you why. If you bet on 1st encounter, you have 1/512 chance to win. If you bet on the 2nd, you have the probability for 1 to not be a shiny + the probability for 2 to be a shiny : 511/512 x 1/512, which is a lower probability than 1/512. And so on... If you bet on the 50th encounter it's (511/512)^49 x 1/512.
With this logic, the highest probability to find a shiny in a hunt is on the first encounter whatever the odds are.