{
  "aliases": [
    "base-rate-neglect",
    "false-positive-paradox"
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  "body": "\n## Example\n\nA disease affects $1\\%$ of a population. A test is positive for $90\\%$ of the sick people and for $9\\%$ of the healthy people.\nLet $S$ be the event \"sick\" and $T$ the event \"positive test\". Then $P(S) = 0.01$, $P(T \\mid S) = 0.9$ and $P(T \\mid S^c) = 0.09$.\nBy the [[law of total probability|0W62TKMW7V]], $P(T) = 0.9 \\cdot 0.01 + 0.09 \\cdot 0.99 = 0.0981$.\nBy [[Bayes' theorem|0NM6E9VW04]], $P(S \\mid T) = 0.009 / 0.0981 \\approx 0.092$.\nSo only about $9\\%$ of the people with a positive test are sick.\nThis is true although the test is right for $90\\%$ of the sick and for $91\\%$ of the healthy.\n\n## Remarks\n\nThe base rate fallacy is the error to ignore the base rate $P(S)$ and to take $P(S \\mid T)$ to be close to $P(T \\mid S)$.\nWhen the condition is rare, most positive results are false positives, even for a good test.\n\n## Explanations\n\n### L1 human\n\nAmong $10000$ people, $100$ are sick and $90$ of them test positive. Of the $9900$ healthy people, $891$ also test positive.\nSo of the $981$ positive tests, only $90$ come from sick people.\n",
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