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".

This is interesting because it shows how tricky the mind can be. I arrived at this web site after reading the book by Kahneman, "Thinking, Fast and Slow". 
I do not see contradiction between the gambler´s fallacy and regression towards the mean. According to the regression principle, the best prediction of the next measure of a random variable is the mean. This is precisely what is assumed when considering that each toss is an independent event; that is, the mean (0.5 probability) is the best prediction. This applies for the next event’s theoretical probability and there is no need for a next bunch of tosses.
The reason we are inclined to think that after a “long” run of repeated outcomes the best prediction is other than the mean value of a random variable has to do with heuristics.  According to Abelson's first law of statistics, "Chance is lumpy". I quote Abelson: "People generally fail to appreciate that occasional long runs of one or the other outcome are a natural feature of random sequences."
Some studies have shown that persons are bad generators of random numbers. When asked to write down a series of chance outcomes, subjects tend to avoid long runs of either outcome. They write sequences that quickly alternate between outcomes. This is so because we expect random outcomes to be "representative" of the process that generates them (a problem related to heuristics). Therefore, assuming that the best prediction for a tenth toss in you example should be other than 0.5, is a consequence of what unconsciously (“fast thinking”) we want to be represented in our sample. Fool gamblers are bad samplers.
Alfredo Hernandez 
