/sci/ - Science & Math

>>10629801

> If you were to randomly generate an infinite list of numbers where every natural number has a 6% chance of being in that list,
then you can find consecutive sequences of any length in that list.

> So your conjecture doesn't really say anything about the clustering you observed.

This is not quite right. You're basically imagining that God is flipping weighted coins, each of which has a 6% chance of getting heads. If he does that long enough then sure, he'll end up with a sequence of heads as long as he wants.

But in reality, we see *far more* sequences of consecutive heads then we should just given the numerical proportion of heads in the integers. Here is a calculator to find the probability of a string of n heads within k trials, given a probability p on each coin: https://www.gregegan.net/QUARANTINE/Runs/Runs.html

Now you can't input a *probability*, exactly--rather you need to choose the nearest integer n to the probability such that n/100 is closest to your probability. 1/16 is 6.25%, which is close enough for now.

We know there is a quintuplet under 1000. If adecous numbers were distributed randomly among the integers, we would expect this to have a 0.08% chance of occurring. We would not expect a quintuplet to appear on average until the upper bound was 1,118,480.

Again: there's a 10-sequence at 15276. The chances of finding a 10-sequence under 20000 is 1.7 in a hundred million; on average you'd need to look at the first trillion-plus integers before finding a 10-sequence.

So it's clear that the distribution isn't random, and that sequences are appearing far more frequently and earlier than they "should". This raises a new question:

Given a sequence length n, is it possible to determine an upper bound below which a sequence of that length (weak version: can be a subsequence, strong version: must be proper) will appear?