Bob,
I'm going to disagree, at least a little. If your dealt royal score is below average AND your total number of royals is below average, your results are going to be below average. But, if your dealt royals are below average and your total royals are about average, your results should be about average.
Let's go with your example of 2 million dealt hands. If this is on 5 play ( the game in question) you will have played 10 million total hands. Using 40,000 for the royal cycle and 650,000 for the dealt royal cycle, you would average 250 total royals and 3 dealt royals.
I content that if you still get about 250 royals and 10 come from dealt royals or 20 come from dealt royals, your results will be about the same.
We might be at the semantic level but there is a point I want to make and this is just because you come up short on one rare statistic of game, does not mean the game is unfair or that your results will necessarily be bad. It would take a lot of work to run a big enough sample to confirm this, especially since Winpoker doesn't track dealt royals. I don't think any other commercial software does either.
Back to what started this discussion, having 2 dealt royals in 5 years doesn't really tell you much about your gaming results for those 5 years. In the 2 million dealt hand example, you will have played 10,000,000 hands and run 50,000,000 coins through. Each dealt royal is 20,000 coins or 0.04% of your return for the 2,000,000 dealt hand run.
In 2,000,000 deals, here are the probabilities of dealt royals:
P(0) = 4.6%
P(1) = 14.2%
P(2) = 21.8%
P(3)= 22.4%
P(4)= 17.2%
P(5)= 10.6%
P(6)= 5.4%
None of those results would be particularly surprising and each dealt royal is 20,000 coins so you would notice it. But as mentioned above, each dealt royal is 0.04% of your return ( I assumed a 100% payback game to make the math simpler).
Interesting discussion.
