This article is very interesting and informative, however it's a bit ironic that an article about misinterpretations of the meaning of the p-value, misinterprets the misinterpretation; in the first blue box it's clear that Bernstein is interpreting the p-value as the probability of randomly rejecting the null (which is what you do when you get something statistically significant) yet in the text following that they say he's interpreting it as the probability of the null. Bernsteins mistake is that he appears to interpret it as an unconditional probability rather than a conditional one (correct interpretation; p-value = Prob(rejecting the null when the null is true)).