Distance between consecutive primes distribution In several prime generation schemes I saw we pick a random number uniformly at random from a wide range and find the next prime after it.

Obviously with such a scheme some primes are more likely than others, a prime is as likely as the distance to the previous prime.

The question is how unbalanced is this? If a friendly alien gave me a list of the top X most likely primes from a much wider range how likely would a random prime from the scheme above be on the list?
First some empirical results, 
over the first 1 million primes the to 0.1% will show up 0.65% of the time
the top 1% will occur 4.7% of time and the top 10% will happen 29% of the time. 

Looking at visual from https://en.wikipedia.org/wiki/Prime_gap

It doesn't look like it has a heavy tail and seems to decline super linearly. this suggest a small tail will never be a large portion. 

For larger numbers the largest ratio between the prime gap and $ln(p)$ is 41 (This ratio is called merit), this is largest known and not an upper bound, but if we take it as an upper bound, and if we take ln(p) as the average distance and 2 as the minimum distance. The most extreme imbalance would be if all gaps were ever maximal or minimal (obviously not the case just for a bound). We mark the ratio of maximal gaps as r. So we will get $ln(p)=r * ln(p)*41+ 2*(1-r)$ 
which leads to $r=(ln(p)-2)/(41*ln(p)-2)$ for $p=2^{1024}$ this comes out 2.5%, and these would come up 99% of the time. But obviously we can't have many gaps of size 2. We can improve this somewhat but not much. 

So we get a huge discrepancy between what we expect and we can prove. If we want a stronger claim then assuming maximal merit things get worse we can bound the gap to be no more than $p^{0.525}$ and if again assume to have a range where gaps are almost entirely either this maximum or very small, we will be able to get a more extreme ratio. Yet even this scenario won't get enough for e.g efficient factorization. If we have a range from $2^{1023}$ to $2^{1024}$ even with maximal gap there are still a lot of primes there. If we would try to factor with trial division we would get no more than a square root speed up over naive trial division which is not competitive with other methods.