(Couldn't get back to this earlier -- busybusybusy before taking a holiday.)

Good body, strange intro - what you're doing in the article is using probability theory to compare a certain statistical tool, correlation, with a different sort of evidence that is much more highly correlated, namely what you're calling "terrible measurement". You're using the probability-theoretic tool of conditional probability and mutual information of probability distributions to point this out.

To present this as a refutation of probability is just odd.

To read it as one is odd. The "strange" intro listed several areas in which probability and statistics are useful (although with slight caveats to the cases of medical research and quality control). The rest is an illustration of its limitations in practice.

I expand more on this in my response to Douglas_Knight and gjm.