[igiveup]

Isn't this just hypothesis testing? [1]

Zero hypothesis: N deaths happen by chance, given a known probability distribution of patient deaths.

Alternative: There was a different probability distribution in play (apparently facilitated by a specific nurse).

P-value: 1 in 342 million, really convincing.

So, is the fallacy that somebody calls the number "probability" rather than "p-value"? Or am I getting it wrong?

[1] https://en.wikipedia.org/wiki/Statistical_hypothesis_testing

[ivanbakel]

No, the fallacy comes from choosing the sample before analysing the probability.

The probability a specific nurse could have such specific bad luck is very low, but there are of course many nurses, and each nurse treats many patients. What is the probability any nurse would have such bad luck, over a long period? How does that probability compare to the probability of murder, which is also estimable? Only either unlucky nurses or murderers end up in the docket - so the p-value really depends on the probability that the prosecutor faces an unlucky nurse versus a murderer.

A simpler comparison: a die with a thousand faces is quite unlikely to land on any particular face. When you roll it, it gives you a sample - is it more likely that the die is weighted to that face, or that the die is fair?

[josephcsible]

This fallacy reminds me of https://en.wikipedia.org/wiki/Testing_hypotheses_suggested_b... in particular. If you didn't have any reason to look for wrongdoing other than a statistical dataset, that same dataset is never sufficient to confirm the resulting suspicion.

[igiveup]

I see. Physicists face this problem with the Large Hadron Collider, and many possible hypotheses explaining its results.

Yet, I think many many nurses are needed to beat the 342 million.

[sealeck]

No the issue is when prosecutors mix up Pr[evidence | innocence] with Pr[innocence | evidence]. It isn't correct to conclude from the former that someone is guilty.

[igiveup]

I believe Pr[evidence | innocence] is p-value (or maybe one minus p-value, not sure). Statisticians use this routinely to test "innocence". It does not mean probability of innocence, but it means something.