Re: Bitcoin puzzle transaction ~32 BTC prize to who solves it

~ compound science ~

Listen, you are correct. Having more prefixes found, in N attempts, after you found a bunch is less likely. So you are entitled to say there will be less found, in N attempts.

I have an upgrade for your theory, please tell me where I have it wrong.

Instead of looking at the range and thinking "hey, less chances of it being here", let's replace that range with some set (just replace the keys of the range with some other keys).

Now, you have less chances of finding more keys in the remaining keys in the set.

So what happens? Well, since you dismiss the rest of the set, you end up going with the continuation of the range, as the "best" new keys that have better chances of finding the prefix.

Do you see the problem now? It's a zero sum game. But my upgrade makes things a lot faster. All good now? It works in perfect accordance with your theory.

Alice flips 100,000 coins, and Bob wants to probabilistically determine how many coins (or keypresses) it takes for 5 consecutive heads to repeat, starting from when the first 5 heads are found. Since Alice's coin flips are immutable (similar to Bitcoin), even though the events are independent, Bob can work with compound probabilities within sequences of flips (ranges).

You haven't answered my question - did I do something wrong with my upgrade? If so, what?

You're revolving in a circle. Not to mention you're using prob. backwards, but that is no longer relevant. Even with your theories, it is still the same thing whether you scan keys in sequence or not.

No, when Alice flips the 100,000 coins, the sequence is immutable, so it is not the same if I calculate the probabilities by randomly selecting keys or flips. The events are already fixed (similar to the concept of immutability in Bitcoin hashes, which have a fixed position). This implies that the way probabilities are calculated is not the same depending on whether you select randomly or follow a sequence. In a case where events can change, I would agree with you.

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.