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bentarm comments on Simpson's Paradox - Less Wrong
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Does anyone know an example where the result is intuitive to most people? The sort of thing where you tell someone the Monty Hall problem with a million doors before they think about the case with 3.
I would guess it would be something with an obvious confounding variable that you've forgotten to factor out. e.g. the "fluoride causes false teeth" example is pretty good for that, but is maybe not quite intuitive enough for what you're looking for.