Law of Large Numbers: $$$ edition

In my field, mean success rates for grant proposals range from a surprisingly generous 30% or so for a rare few opportunities, to a more typical 6-10%

As the number of proposals submitted go to infinity, most all researchers must have near average success rate.

I was prodded by this in noting that I was on a fair number of Hubble proposals, and my hit rate was exactly the average.

In the limit of random funding, this is of course exactly what happens, the average researcher will have a long term success rate for funding which approaches the mean success rate as the number of proposals goes to infinity.

Now, I don't think the distribution of research success is actually gaussian, clearly there is is a power law tail, with a larger than expected number of "overly successful" researchers, who have consistent above average success rate (at least for a finite number of proposals submitted); conversely there are also clearly researchers with consistently lower than mean success rate.

But, necessarily, for most researchers the long term grant success rate must approach the mean for the field (I guess I am arguing that the mode is not and cannot be significantly offset from the mean).

So don't think you're special just because you got a grant!
In the long run you will have to submit 2-15 grants that fail to make up for it... (depending on what panels you're submitting to, and assuming no secular trend leading to monotonically increasing success rates)

So there.