Regression towards the mean v/s the Gambler's fallacy Suppose you toss a (fair) coin 9 times, and get heads on all of them. Wouldn't the probability of getting a tails increase from 50/50 due to regression towards the mean?
I know that that shouldn't happen, as the tosses are independent event. However, it seems to go against the idea of "things evening out".

The Gambler's Fallacy is the incorrect belief that after a sequence of random events of one kind, the next event is more likely to be of an opposite or different kind. In the case of an equilibrium coin toss, odds of the next event are the same as the previous event, that is of equal chance. That is not inconsistent with our sense that things tend to even out over time as long as we appreciate that time is not defined as a single event but as a sequence. 
A coin toss involves random chance the result of which cannot be determined but only described probabilistically as 50/50 per event and as something tending toward a mean of 50/50 in an indefinite sequence of such events.
