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can you give examples of how statistics violates common sense?

A classic example would be that if you have a test (say for cancer) with a false positive rate of “””only””” 5%, and your disease has an incidence of say 1 in 1000.

Let’s say that you get a positive diagnosis for the disease, and you ask someone the question:

What is the probability you actually have the disease?

Most people will say 95% or 99%, but your actual probability of having the disease in this example is <2%

Unfortunately that well-worn example usually only proves that "false positive" as a technical term fails to match people's intuitions. The underlying problem about the base rate is important to teach, but it's easy for well-meaning people to try and teach the base rate lesson but fail by instead teaching a bullshit gotcha about the definition of "false positive."

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