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