A trace left at a scene is traced to a town of 20 000 people…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A trace left at a scene is traced to a town of people. Exactly one person in the town is the true source of the trace, and that person always matches the profile. Any other person matches by chance with probability . A suspect is chosen from the town at random, with no other evidence, and the suspect's profile matches. A newspaper writes that the suspect must be the source because the chance of an innocent match is only . A statistician replies that this is the prosecutor's fallacy. Find the probability of a chance match for one person who is not the source. Find the expected number of other people in the town who match by chance. Then find the probability that the suspect is the source, given the match. Give the last answer to four places.
Need a reminder? The key ideaShow
A test's sensitivity is , and its specificity is . What a result means depends on the base rate. List the sample space, cross out what the news rules out, and use Bayes' theorem to reverse a condition.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. A chance match. .
Step 2. Others who match. There are people who are not the source. The expected number of chance matches is , which is about .
Step 3. The prior. The suspect is the source with probability .
Step 4. The posterior. The source matches for sure. A non-source matches with probability . So
Answer: , about other people and about .
The fallacy. The newspaper treated as if it were . The right value of the second is about , which is far from .
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