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?