Edit: I would appreciate any kind of input.
This is an old homework task I'm still not able to solve:
A DNA sample was taken from a murder scene. Statistically, only 1 in a million people is a match. However, the test used is erroneous in $0.001\%$ of the cases. One of $20$ tested suspects tests positively. What is the probability that the one with the positive test result is not guilty (the actual source of the sample)?
Additional conditions: The suspects are the only possible murderers. The source of the DNA is in fact the murderer, and no one else was involved in the murder.
So using Bayes' theorem, the probability of not being a match despite a positive test result in general is
$$P(¬M|+)=\frac{P(¬M)\cdot P(+|¬M)}{P(+)}$$
$$=\frac{\frac{999999}{1000000}\cdot \frac{1}{100000}}{\frac{999999}{1000000}\cdot \frac{1}{100000}+\frac{1}{1000000}\cdot\frac{99999}{100000}}=\frac{111111}{122222}\approx 0.90909$$
Now how do I proceed? Can I calculate the probability for one of the others to test negatively despite being a match (the source), and then calculate the probability for the remaining 18 to test correctly (negative), and then multiply everything together? The result is tiny, which at least makes sense intuitively.