An investigative bureau uses a laboratory method to match the lead in a bullet f

Question

An investigative bureau uses a laboratory method to match the lead in a bullet found at a crime scene with unexpended lead cartridges found in the possession of a suspect. The value of this evidence depends on the chance of a false positive positive that is the probability that the bureau finds a match given that the lead at the crime scene and the lead in the possession of the suspect are actually from two differant melts or sources. To estimate the false positive rate the bureau collected 1851 bullets that the agency was confident all came from differant melts. The using its established ctireria the bureau examined every possible pair of bullets and found 658 matches. Use this info to to compute the chance of a false positive.

Explanation / Answer

To answer this question we need to knwo something about the binomial distribution. Here we have 1851 bullets that we KNOW are NOT MATCHES of one another. One by one they examine two bullets at a time. This is where the binomial distribution comes in: there are 1851 bullets but each time we choose two. For more discussion on the binomial distribution, see here: http://en.wikipedia.org/wiki/Binomial_distribution Specifically, we are using the probability mass function. We say "of N, choose K" which corresponds to "of 1851 bullets, choose 2", and we want to know how many ways we can come up with unique pairs of bullets. This is analogous to how many tests the FBI is running in this case. Using N choose k = N!/k!(N-k)!, we have 1851 choose 2 = 1851!/2!(1851 - 2)! = 1851!/2*1849! = 1850*1851/2 = 1,712,175 possible combinations. Of these combinations, there were 658 false matches. So the odds of getting a false positive is 658 / 1,712,175 = 3.84 x 10^-4, or 0.0384% Hope that helps! Jonathan

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