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RobinZ comments on Beautiful Probability - Less Wrong
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I think you're misunderstanding probability theory a little. Probability theory is the chance that x result means that the universe operates on principle y. The fact that the second researcher had no reason to stop until he tested 100 people means that the % of patients cured did not exceed 60% until his last batch of test subjects. Which significantly alters the chance that the universe operates on principle y. The first researcher could have had a % over 60 at any time during the experiment. Which is a physical difference. The probability used is not based on the researcher's private thoughts, it is based on their experimental procedure, which is different, regardless of the fact they ended with the same results.
First off: welcome to Less Wrong! I hope you're enjoying the articles you've read so far.
That said: I don't think you really address the thrust of this article, here. The point is that all we have learned about the principle the universe operates on from either experiment is that said principle produced a given result - the exact same result, by stipulation. Any difference in the conclusions drawn from that result must, therefore, arise from difference in assumptions about the universe - because the two experimenters are working from literally the same data.