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Suppose there has been a hit-and-run accident. A person was struck by a taxicab last night. There are two cab companies in town, Blue Cab and Green Cab. The only witness, who was some distance away, believes it was a blue vehicle he saw cause the accident. Which cab company is guilty? How much do we know?

How would it affect your conclusion if you also knew:

- Green Cab Company had 90 cabs on the street at the time of the accident, while Blue Cab Company had only 10 cabs out.
- Given the lighting and visibility at the time of the accident, a person could correctly distinguish a blue from a green cab correctly about 80% of the time, at the distance the witness said he was from the accident. (One can always tell if it was a taxicab or not.)

Now we have a "prior probability" that the cab was blue: 10%. (This is the probability that the cab was blue even if there were no further color information from witnesses.)

We can also estimate the quality of the evidence from the witness: If it was a blue cab, the witness should have been able to identify the color correctly with 80% probability. If it was actually a green cab, the witness would have said it was blue with 20% probability (the probability of a witness making an error in color identification).

The calculated probability cab was blue, given testimony it was blue = (probability the witness would testify the cab was blue, given that it actually was blue (80%) × prior probability cab was blue (10%)) ÷ probability the testimony would be that the cab was blue (26%).

(The witness would testify the cab was blue 80% of the time if it was blue (of which there is 10% chance) and 20% of the time if it was green (of which there is a 90% chance) = 26% of the time overall.)

So what is the calculated probability the cab actually was from Blue Cab Company, given all of this information, and using Bayes' Theorem? Just 30%!

Given this testimony and these facts, there is a much higher probability that the cab involved in the accident was actually from Green Cab Company (70%)!

Why, you may ask, is any of this Bayesian stuff relevant?

Try substituting "black person" for "Blue Cab", "white person" for "Green Cab", "mugging" for "accident", and "I think I saw a black person running away" for the testimony of the witness, and then imagine yourself on the jury. Any jury member would thoughtfully consider how accurate the witness's testimony was likely to be, considering the conditions and other factors. But now you know how important it is to also consider the prior probability that the assailant was black, in the absence of any evidence one way or the other.

Interestingly, some courts have held that juries don't have to consider Bayesian probabilities, but can just use their (so often mistaken) intuition.

If you are interested in how judges have tried to understand Bayes' Theorem, check these sites:

http://www.law.umich.edu/thayer/redmay.htm

http://www.mcs.vuw.ac.nz/~vignaux/docs/logicalJuries.html

Remember, one of the reasons those judges went into law in the first place is that they heard they wouldn't have to do any math.

This page supports an article on Bayes' Theorem at the main Science In Action site. Click here to return to that article, and here to go to the Science in Action home page.

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