Vladimir,
Thanks for doing that. I guess I was doing it wrong using Binomial Distribution when I should have been using Poisson. My understanding is that Binomial Distribution is based on a fixed set of trials whereas Poisson is based independent events occurring over time, which is the correct thing to use for finding bitcoin blocks.
So my 0.0036 (277:1) figure was incorrect it should have been
0.00695 (144:1) as you said.
And either way, the best way to catch it would be to look at the average over the longest term possible. You can always find bad days. Half the weeks will be below average. And if you get to pick the start date and the finish date specifically to bracket a run of bad luck, you can always find improbable events.
I want to know if the pool is being bounced up and down on purpose (with the down days likely being a little more down than the up days are up) to try to make it more difficult to see what is going on.
So eleuthria says that during difficulty 1563027 there is a 90% chance the number of blocks found should have been higher so 10% chance they were this low by chance. That's because that +70% and other high luck days that followed the low days made up for a lot of it.
If some more positive luck days are added to the pool it could be made to look just fine without any missing blocks problems at all. But the low days and high days will remain there as evidence that manipulation -- such as stealing and then a cover up -- took place.Not very many bitcoins were stolen over all so far. In fact the estimated amount of bitcoins stolen could go to 0 in the future if more positive luck days are added to make up for it.