What is the probability of John being guilty given a positive DNA match?

[fhjop1]

John has been accused of murder and the police have found DNA
evidence at the scene. The probability of a DNA match given that a person is
innocent is 1/100,000. The probability of a DNA match given that a person is
guilty is 1. John lives in a city where there are 100,000 people who could have
committed the crime. Unfortunately, the outcome of the DNA match to John has
been positive. What is the probability of John being guilty given the outcome of
the DNA test?

I don't understand that if the probability that a DNA matched given a person guity is 1, how come he/she can be innocent, why the 1/100000 can still happen. And does anybody think 'the 100000 people in John's city' this condition is useless? anyway, I think the answer is 99999/100000, cause ignore whether John is guilty, he still have 1/100000 probability can be innocent, is it right?

 

[micromass]

The problem says that if a person is guilty, then there will be a guaranteed DNA match (because the probability is 1).
It does not say that if there is a DNA match, then the person is guilty! You seem to assume this. It can very well happen that a person is innocent and still gets a DNA match.

For example, it is true that if I fall then I will hurt myself. But it is not necessarily true that if I hurt myself, that I must have fallen. There are other ways to hurt myself.

 

[Ray Vickson]

John has been accused of murder and the police have found DNA
evidence at the scene. The probability of a DNA match given that a person is
innocent is 1/100,000. The probability of a DNA match given that a person is
guilty is 1. John lives in a city where there are 100,000 people who could have
committed the crime. Unfortunately, the outcome of the DNA match to John has
been positive. What is the probability of John being guilty given the outcome of
the DNA test?

I don't understand that if the probability that a DNA matched given a person guity is 1, how come he/she can be innocent, why the 1/100000 can still happen. And does anybody think 'the 100000 people in John's city' this condition is useless? anyway, I think the answer is 99999/100000, cause ignore whether John is guilty, he still have 1/100000 probability can be innocent, is it right?

Some innocent people may match the DNA profile, because the city has 100,000 people and the chance that any random innocent person matches the profile is 1/100,000. The number of innocent matches is a Poisson random variable with mean 1.

RGV

 

[Curious3141]

John has been accused of murder and the police have found DNA
evidence at the scene. The probability of a DNA match given that a person is
innocent is 1/100,000. The probability of a DNA match given that a person is
guilty is 1. John lives in a city where there are 100,000 people who could have
committed the crime. Unfortunately, the outcome of the DNA match to John has
been positive. What is the probability of John being guilty given the outcome of
the DNA test?

I don't understand that if the probability that a DNA matched given a person guity is 1, how come he/she can be innocent, why the 1/100000 can still happen. And does anybody think 'the 100000 people in John's city' this condition is useless? anyway, I think the answer is 99999/100000, cause ignore whether John is guilty, he still have 1/100000 probability can be innocent, is it right?

Let G mean guilty, and M mean DNA match. (The complementary outcomes are NG, not guilty and NM, non-match).

You're given that the probability of a match given guilt is 1, i.e. $p(M|G) = 1$

You're asked to find the probability of guilt given a match, i.e. $p(G|M)$. This is NOT, in general the same as the above.

This question can be solved using Bayes' theorem. Do you know it?