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The Prosecutor's Fallacy

Bayes' Theorem ... A Simple Example

Notation: Prob(A) means "the probability of event A" and Prob(A|B) is "the probability of event A, given that event B has happened."

Bayes' Theorem: Prob(A|B)xProb(B) = Prob(B|A)xProb(A)

Now, Prob(A|B) and Prob(B|A) are often confused by even the most intelligent of people. The confusion often appears in legal cases and is sometimes called the Prosecutor's Fallacy. Bayes' Theorem relates these two distinct conditional probabilities.

Followed by a straightforward example of why this really matters.

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If it is known that the pool of suspects is 25 ppl, and that the suspect must have red hair, can't you redefine the pool of suspects to 2? That's what I want to do...

Yes. That is what the author implicitly said with "The innocent redhead gets a better deal if it is reported that the probability that a person with red hair is innocent is 50%."

As the author also noted, the example is really too simple. But it leads into the comments at the bottom about a more realistic situation: "The prosecutor tells the jury that the probability an innocent person has red hair is .0099, or 0.99%."

Which sounds impressive until you notice that only about 1% of the whole population is red-headed.

All of this sounds rather obvious, but this sort of mistake, or misrepresentation, really does happen.

Edited at 2013-04-08 08:01 pm (UTC)

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