Secondly, I'd like to clarify a few things. I recognize that practice is an extremely important factor in improving, I have never denied this. Players like Tommy and [name removed]are much, much better than I am, and in very large part this is due to an awful lot of hard work on their part. I do not expect that lucky mode would be a short cut to improving my times, and I do not expect that it could magically allow me to beat my records in a week or compete with elite players. And I am not getting hung up on individual games lost to 50-50s. I can't even bring to mind an occasion when I have missed out on a good time, and I would like to think that if I blasted say a 43 or something that I would be philosophical about it and treat it as a motivation like Tommy suggests. But what is happening is that I am getting a tiny bit frustrated

*thousands*of times because of things we know how to avoid, and there is a cumulative effect.

You keep saying that the benefit to completion rate of lucky mode depends on the fraction of boards you blast through mistakes like misclicks, but I don't think that is true. Unless you believe the probability that you will make a misclick or other mistake on a board with no 50-50s is different from the probability on a board with 1 or more 50-50s (and I see no reason to believe there would be any significant difference), then the two effects are independent. For example, assume you have a probability p of blasting through a misclick and consider a distribution where 40% of boards have 0 50-50s and 60% have exactly 1. (For reasons of simplicity I have assumed a single probability p, but it would work equally well with a distribution, and I have absorbed multiple 50-50s into a larger probability for a single 50-50, but the analysis would be much the same with the full distribution of 50-50 prevalences that qq determined). So, the total fraction of boards you blast will be

( p * fraction of 0 50-50 boards ) + ( p * fraction of boards with 1 50-50 which you guess correctly if you get to it) + ( 1 * fraction of boards with 1 50-50 which you guess incorrectly if you get to it)

= 0.4 p + 0.3 p + 0.3

So, the total fraction of boards blasted is

0.7 p + 0.3

Or, the fraction of boards completed is

1 - ( 0.7 p + 0.3 )

= 0.7 - 0.7 p

= 0.7 ( 1 - p )

If the guesses were removed, then you would complete a fraction 1 - p.

So the increase in completion rate would be

( 1 - p )

_______ = 1 / 0.7

0.7 * ( 1 - p )

a result which is independent of p.