Perhaps to explain independent probabilities a little bit better, let's talk about averages, tails, and expected times. For simplicity, let's call the inverse of a chance an encounter is shiny with the letter x (i.e. x = 8192). For no other reason than it's shorter to write.

On average, you will encounter one shiny every x encounters. They come randomly so you won't get them exactly when you think you should. You might not get them at all. But on average, over everyone playing Pokemon, the total encounters divided by the total shines will be about x.

However, because all of these events are independent, if you don't get a shiny in, let's say, 6x encounters, that doesn't change anything at all. From this point on you're still only expecting a shiny about every x encounters. The universe doesn't increase the rate just because you previously got bad luck. Similarly, if you got 6 shinies in x encounters, you're still only expecting a shiny once about every x encounters.

The key takeaway is that shiny hunting doesn't remember the past. It has no history. That's both good and bad, because a lucky streak cannot harm you, but an unlucky streak cannot help you.

Finally, let's talk about the long tail.

Let's number your attempts. Attempt #1, attempt #2, and so on. And let's say you only want to catch one shiny, so you stop after finding it. Which attempt number do you have the greatest chance of catching that shiny?

Believe it or not, the answer is the Very. First. Attempt.

Let's see why.

The probability that any particular pokemon is shiny is 1/x, let's call this y (i.e, y = 1/8192). So the first pokemon being shiny has probability y. If you get to the second pokemon, it also has a probability of y, but that's only if you get there, which has a probability of (1 - y), so the total probability that you catch a shiny on your second encounter is y (1 - y). Attempt #3 is y(1 - y)^(2), and so on.

Let's say you tried to catch a shiny 200000 times. Which attempt has the highest chance of having the shiny? The very next attempt, of course, with probability y. Getting to this attempt without a shiny was indeed quite unlucky, but now that the past has already happened, the 200,000th attempt and the 1st attempt are the exact same.

That's what the other user was getting at with the 10 hour session. If you play for 10 hours you have a certain chance of getting a shiny. If you play for 1 hour you have a different chance. But the idea is that if you decide to play for 10 hours and you don't get a shiny in the first 9, then your chance of getting a shiny in the last hour is the same as your chance of getting a shiny in only 1 hour, rather than your original 10 hour chance.

Hope this helps.