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Suppose that the FBI database has fingerprints of 10 millon people and the probability that two fingerprints being falsely matched is 1 in 5 million. If someone's fingerprint is in the database, it will certainly come up if he or she commits a crime.

Let $F$ denote that the criminal’s fingerprint is in the database and $F^c$ not in the database. $X$ be the number of matches found in the FBI database when the police run the crime-scene sample.

Find

a) $P(X=1|F)$

b) $P(X=1|F^c)$

c) $P(F^c|X=1)$


My thoughts:

a) Since it is given that a match will come up if the fingerprint is in the database, $P(X=1|F)$ should be 1. But again, should we consider the case that when another record falsely matches, and $X$ could be more than 1. I am kind of confused here.

b) Can we use bayes' theorem to do this part?

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    @John.Mathew: Just out of curiosity, do you actually work for the FBI?2011-11-12

1 Answers 1

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Since this is homework, I won't give full answers, but these hints should get you there.

a) We're told that we'll get one match from the actual fingerprint; for $X$ to be 1 we require that the other 9999999 fingerprints don't match. They're independent events.

b) This time we have no guaranteed matches. We require one false match and 9999999 non-matches from 10000000 events - but make sure you take into account that the false match could be any of the events.

c) This is where you can use Bayes' theorem.