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Intro: p-values
Originally: to indicate a problem with a model worth investigating. Not: p small therefore other model correct. 
p-values are not probabilities
p-values are random variables between 0 and 1

So the p-values should 

The exact value of p-values below 1e-6 is unreliable

Effect size may be more interesting
http://astrost.at/istics/minimal-statistics-checklist.html