Again, if it’s guaranteed to be mathematically exactly fair, then by the maths I posted earlier, claiming you have better than 1/6 chance of getting the next one right is mathematically impossible, by definition.

To be clear: you’re defining a situation whereby you are guaranteed to only have a 1/6 chance of getting the next number correct, whichever you pick, and then saying “isn’t it better to stick with the number that came up before?”. Simply, no, it’s not, because of the way you defined the system.

If you’re trying to construct an actual scenario, a casino wouldn’t let that happen. They’d kick the player out because “they believe them to be an advantaged player”, because they don’t like losing money. And eventually you reach a point where it’s simply more likely that there is a bias somewhere in the system that hasn’t been detected yet.

That means it would be a feature of the system (player / table / dice) rather than of the maths - the maths is based on perfectly controlled probabilities.

Practically, you can’t ensure it’s a 100% fair system, so the simple “each outcome is 1/6” breaks down. If you could guarantee that it was perfectly fair, then what I said earlier stands. In a Real-World situation, the assumptions change significantly - you can’t have perfect knowledge of everyone’s intentions, whether it could be a scam etc.

EDIT: however, most gamblers fallacies aren’t based on the idea “I have actual evidence that the system is rigged”. Things like “5 hasn’t come up on the roulette wheel, it must be overdue” aren’t based on an assumption of bias, they’re based on an assumption of fairness, which says that eventually all numbers will come up equally. However, they don’t have to come up equally before the heat death of the universe.

Ultimately, it doesn’t matter which number I pick, I’ve still got the same chance of being right as with any other number.

People can use superstition to help them decide, but it doesn’t make them any more likely to be right. Some people will choose 6 because “that’s got to be right, it’s happened so often”. Others will choose their favourite number “because that’s lucky”. Others will choose anything but 6 because “they can’t be that lucky”. Any logic you try and apply to it to say “this outcome is more likely than any other” is just your brain tricking you.

The “absurdly unlikely” part comes in to play in being able to view the events from two different perspectives.

One is that, if someone claimed at the outset, that they could predict the next quintillion rolls of the die (whatever values those might be), the probability of all of them being correct is vanishingly small - each of the 6^quintillion combinations almost certainly won’t show up, only one will, and you’re relying on picking that one sequence.

However, once you’ve correctly predicted the quintillion rolls in a row, if you then say “I’m going to roll a 6 next”, you aren’t any more or less likely to get it right than you were on the first roll.

The probability of being able to predict (N+1) correct dice rolls is N * 1/6.

1 roll: 1/6

2 rolls: 1/6 * 1/6 = 1/36

3 rolls: 1/36 * 1/6 = 1/216 Etc.

If you’ve already done the N dice rolls, you’ve already dealt with the probability of getting to where you are in the chain of rolls. The probability to advance to the next step in the chain is always the same though, even if the chances of you successfully getting to that point in the chain are infinitesimal. You’d still expect 5/6 to get it wrong at the next roll, 35/36 to get it wrong in the next 2 rolls, and 215/216 to get it wrong in the next 3 rolls.