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RE: A common error regarding the p-value
Yes, and no.
From a strict mathematical perspective, you're right, of course. I've always found the precise meaning of p to be a little hard to articulate in a way that really conveys the difference, though, and unless it's relevant, I don't bother.
In general, on the additional hypothesis that the experiment is correctly constructed, the p value does become the probability the tested effect occurred by chance. In a correctly constructed experiment, there are only two possible causes to which one can attribute an observed effect - to "chance" or to the treatment under test.
So, p (properly) is the probability the observed difference would occur if the treatment was ineffective. But, our experiment is constructed so that if the treatment is ineffective, then the observed outcome is due to chance. So, under those conditions, p is the probability that the observed difference would occur due to chance.
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