I don't see the illusion.. I believe the past events should be entirely discarded. So whether it was 1.9 or 2.1 or a 100, it shouldn't matter.
Well, you're right that they should be discarded and that they don't matter. But it's easy to see why people think it does matter:
1) so far we've seen 1 head and 9 tails
2) we know that in the long run we will see roughly the same number of each
3) so therefore we must be 'due' to see more heads than tails
1 and 2 are true, and naively step 3 seems like a natural consequence of 1 and 2. That is where the error happens. Step 2 happens in spite of step 1, because "in the long run" the difference in step 1 becomes insignificant. There's no tendency to reduce the difference, only to make it not matter.
If there are two 1% house edge sites with same bankroll at the point, one with -1% (past) actual return on wagered and another with 2% actual return, and you are to invest in one of them, I wouldn't see any difference between the two.
You need to take into account that one of the sites may be cheating its investors. If everything else was the same I think it would make sense to invest with the more profitable one, since they are less likely to be cheating their investors. But this (much like Twitchy's comment that maybe the coin has tails on both sides) is kind of beside the point. For the purposes of this discuss we should assume a fair coin, and a fair gambling site.
