Why "only 1/10,000 men with wives they abuse subsequently murder them" ≠ P(A|G,M) & "50% of husbands who murder their wives abused them” ≠ P(G|A)?
All emboldings are mine. See my red side line — the solution identifies "only 1 in 10,000 men with wives they abuse subsequently murder their wives" (in the problem statement) with $\color{red}P(G|A)$.
See my green underline — the solution identifies "50% of husbands who murder their wives previously abused them" with $\color{limegreen}P(A|G,M)$.
But why not vice versa? Why doesn't "only 1 in 10,000 men with wives they abuse subsequently murder their wives" correspond to $\color{limegreen}P(A|G,M)$, and "50% of husbands who murder their wives previously abused them" $\color{red}P(G|A)$?
☣ 2.8.2 (Defense attorney's fallacy). A woman has been murdered, and her husband is put on trial for this crime. Evidence comes to light that the defendant had a history of abusing his wife. The defense attorney argues that the evidence of abuse should be excluded on grounds of irrelevance, since only 1 in 10,000 men with wives they abuse subsequently murder their wives. Should the judge grant the defense attorney's motion to bar this evidence from trial?
Suppose that the defense attorney's 1-in-10,000 figure is correct, and further assume the following for a relevant population of husbands and wives: 1 in 10 husbands abuse their wives, 1 in 5 murdered wives were murdered by their husbands, and 50% of husbands who murder their wives previously abused them. Also, assume that if the husband of a murdered wife is not guilty of the murder, then the probability that he abused his wife reverts to the unconditional probability of abuse.
How to define the "relevant population" and how to estimate such probabilities are difficult issues. For example, should we look at citywide, statewide, national, or international statistics? How should we account for unreported abuse and unsolved murders? What if murder rates are changing over time? For this problem, assume that a reasonable choice of the relevant population has been agreed on, and that the stated probabilities are known to be correct.
Blitzstein. Introduction to Probability (2019 2 ed). pp 75-76.
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