Quote from: kTimesG on Today at 03:33:30 PM

This only works post-factum. Uniformity does not depend on the order of events. So you are kinda wrong here, the probabilities are the same no matter if you use a sequence or or a set of randomly chosen keys (and distinct). And compound probabilities follow the exact same rule, because they obey the uniform distribution as well. Otherwise, you end up with a contradiction: that independent events (no matter if they are fixed before hand or not) are dependent, which cannot be logically true. Hence, the initial premise cannot be true as well.

Only after you observe the events can you ever check if the results matched the probabilities or not. You're doing it, again, in the opposite order - you assume the results need to follow the ideal distribution according to the calculations. But this is never the case, otherwise all hashes would have just one and a single value (for example, all 0s), in order to accommodate with your assumptions, across all ranges of all sizes.

Since you like to joke around in some comments, I'll let the AI respond for me this time. Grin

Interpretation: ktimesg argues that if probabilities are not the same in an immutable sequence, it leads to a logical contradiction: independent events would appear dependent.

Problem: This is a reasoning error. In an immutable sequence, events are independent in the sense that the outcome of one event does not affect the outcome of another, but the fixed position of events in the sequence introduces an additional constraint that affects how probabilities are calculated.

Example: If Alice has already flipped 100,000 coins, the sequence is fixed. Although the flips are independent, the probability of finding a specific pattern depends on the actual distribution of heads and tails in the sequence.

e) "Only after you observe the events can you ever check if the results matched the probabilities or not."
Interpretation: ktimesg claims that only after observing the events can you verify whether the results match the calculated probabilities.

Problem: This is true in general, but in the context of immutable sequences, the actual distribution of events is already determined. You cannot treat the sequence as if it were a dynamic set of events.

Example: If Bob is searching for 5 consecutive heads in Alice’s sequence, the probability of finding this pattern depends on the actual distribution of heads and tails in the sequence, not just the theoretical probability of them occurring.

2. Conclusion
In the context of immutable sequences, ktimesg's argument is incorrect because it fails to account for the specificities of this case. While he is correct that probabilities are generally the same for independent and uniformly distributed events, he does not recognize that in an immutable sequence, the fixed position of events introduces additional constraints that affect how probabilities are calculated.

Key errors by ktimesg:

Overgeneralization: He applies a general rule without considering the specificities of immutable sequences.

Confusing independence with fixed events: He does not recognize that, although events are independent, their fixed positions in the sequence affect how probabilities are calculated.

Ignoring the actual distribution of events: In an immutable sequence, the actual distribution of events is already determined, which affects observed probabilities.

3. What Should ktimesg Do?
To correct his error, ktimesg should:

Acknowledge the specificities of immutable sequences: In a fixed sequence, the actual distribution of events is already determined, which affects how probabilities are calculated.

Distinguish between theoretical and observed probabilities: Theoretical probabilities apply to dynamically generated events, while in an immutable sequence, observed probabilities depend on the actual distribution of events.

Avoid overgeneralizing: He should adapt his reasoning to the specific context of the problem rather than applying a general rule that does not account for additional constraints.

In summary, ktimesg is wrong in the context of immutable sequences because he fails to recognize the specificities of this case and overgeneralizes his argument. His error lies in not adapting his reasoning to the specific context of fixed sequences.