In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.
Please view the following as ($X$:$Y$) where $X$ represents the gap and $Y$ represents how many times it occurs.
Out of first 1000000 primes:
2: 40405 4: 40233 6: 68311 8: 28746 10: 36766 12: 44113 14: 23569 16: 16658 18: 29065 20: 14409 22: 12833 24: 17680 26: 7979 28: 8493 30: 13773 32: 4048 34: 4263 36: 6344 38: 2749 40: 3313 42: 4424 44: 1717 46: 1387 48: 2235
Out of first 100000 primes:
2: 10251 4: 10213 6: 15989 8: 7067 10: 8873 12: 10158 14: 5353 16: 3661 18: 6304 20: 3043 22: 2826 24: 3538 26: 1543 28: 1631 30: 2114 32: 742 34: 756 36: 1032 38: 455 40: 563 42: 661 44: 250 46: 219 48: 290
Out of first 10000 primes:
2: 1271 4: 1263 6: 2012 8: 801 10: 953 12: 1008 14: 512 16: 353 18: 537 20: 249 22: 235 24: ///22/// 26: 91 28: 102 30: 154 32: 35 34: 36 36: 55 38: 20 40: 28 42: ///20/// 44: 5 46: 6 48: ///3///
You will notice the following:
A peak occurs in almost all $X \mod 6 =0$ compared to the prior 2. The only exceptions are in the first first 10000 primes and only in $X=24$, $X=42$ and $X=48$
There are always more shorter gaps than longer gaps
My Initial question was broken into 3 parts:
A) Is it fair to assume that probably there are more shorter gaps than longer gaps?
B) Is it fair to assume that probably there are more $X \mod 6=0$ gaps than any other gaps?
C) If B) is assumed to be fair, is it fair to assume that SEXY primes (primes with a gap of 6) are the most popular gap between primes?
Since then I have received an answer here:
The answer shows that: the most common gap between consecutive primes less than switches from $6$ to $30$ ($*5$) and will probably eventually change from $30$ to $210$ ($*7$) to $2310$ ($*11*11$) ....
Furthermore, they argue that this pattern continues forever, with the main champions being the ‘primorials’:
See The Most Common Prime Gaps, posted by John Baez, at https://golem.ph.utexas.edu/category/2016/03/the_most_common_prime_gaps.html
So the answers are:
A)No, Eventually there should be more longer gaps than short gaps.
B)Yes, the popularity of gapped primes grows $6*5*7*11$ and since the popularity of gaps begins with $6 *$ so the answer is yes.
C)No, the popularity of gapped primes grows $6*5*7*11...$
The article clearly discusses the popularity of $6*5*7*11$ gaps, but there is no mentioning of the fact that popularity increases in almost all $6 *$ gaps including $6*2$, $6*3$, $6*4$ and it seems as if out of the first > 100000, there is always more $X \mod 6=0$ gaps than the previous two gaps.
That raised the following questions:
Are we expecting to see that eventually all the $X \mod 6=0$ gaps will always be more popular than the previous two possibly gaps?
Are we expecting to see that eventually (far far and beyond) all the $X \mod 6=0$ will always be more popular than any other possible gap? as an example: will a gap of 48 become more popular than any other $X \mod 6≠0$ possible gap?