When the Better Drug Loses
Drug A succeeds in 80% of mild cases and 30% of severe cases. Drug B succeeds in 70% of mild cases and 20% of severe cases, so A is better in each group. Drug A is given to 20 mild patients and $n$ severe patients; drug B is given to 70 mild and 30 severe patients. What is the smallest integer $n$ for which B's overall success rate is strictly higher than A's?
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Source: Simpson's paradox (Yule 1903, Simpson 1951); numbers original. Statement written for AxiomIQ.