Quote from: mcdouglasx on January 31, 2025, 11:19:57 PM

Quote from: kTimesG on January 31, 2025, 03:33:33 PM

What reference do you have that states compound probability refers to two independent events occurring one after the other and not anywhere in the number of trials.

In the case of compound probability, on the other hand, which refers to the probability that two or more independent events occur in sequence, you could also apply it to Bitcoin by calculating the probability that a prefix repeats as you move from one prefix to another.

But in any case, it cannot be taken as a single independent event, it is not like you are looking for a different prefix in each event. In the same way, this book explains it with examples.

Again, you are using the words "occur in sequence" which is not the same as "occur in a sequence". There's nothing in that book about this.

You're still assuming the probability changes just because you move from one key to the next. But there's no external entity that would do that (except faith maybe).

Example: some sequence of 100 events, and you know somehow that 2 of them are successful ones. It doesn't matter what we mean by "successful" (hash prefix match, or whatever you'd like to use).

Total possible sequence combinations: 4950
Total combinations where successes are next to each other: 99

So, the probability to have the successes next to each other is 99 in 4950.

Total combinations where successes are separated by a distance of 2: 98
Total combinations where successes are separated by a distance of 3: 97
Total combinations where successes are separated by a distance of 4: 96
...
Total combinations where successes are separated by a distance of 50: 50

So, the probability to have the successes separated by a distance of 50 is: 50 in 4950.

At this point, you would be something like: OK, this sounds like it's actually more likely to have the successes next to each other, rather than evenly spaced! So WTF is happening here?

Let's continue:

Total combinations where successes are separated by a distance of 51: 49
Total combinations where successes are separated by a distance of 52: 48
...
Total combinations where successes are separated by a distance of 98: 2
Total combinations where successes are separated by a distance of 99: 1

Maybe now something starts to become obvious: wait, so we only have 1 in 4950 as a probability to have the successes spaced by distance of 99? That doesn't sound right.

Well, friends, what we did is we forgot about what happened before our sequence and after our sequence. So, if we put head to head our sequence of events, we'll see that if the successes are at #1 and #100, they are "next to to each other".

So let's fix our probabilities:

Total combinations where successes are next to each other (or at dist 99): 99 + 1 = 100

So, the probability to have the successes separated by a distance of (1 or 99) is: 100 in 4950.

The same goes for the other ones, let's see what happens:

Total combinations where successes are separated by a distance of X (or 100 - X): 100 - X + X= 100

So, same chances whatever delta we pick.