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Question:

Roughly $1$ in $1{,}000{,}000$ is a murderer. In an ongoing murder investigation a blood sample is taken from the murder scene. A DNA test, which accurately identifies $99$ out of $100$ blood samples, is performed on a randomly selected person. The DNA test falsely identifies $1$ out of $1{,}000$ samples (false positive. The test results in a positive match. What is the probability that this person is the murderer?

My try:

Let $M$ be the event that the person is a murderer a priori DNA test. So $$P(M)=\frac{1}{10^6}$$

Let $D_{acc}$ be the event that the DNA test is accurate and $D_{iacc}$ be the complement of $D_{acc}$, that is the probability that the test is inaccurate. So we have $$P(D_{acc})=\frac{99}{100}$$ and $$P(D_{iacc})=\frac{1}{100}$$ Let $D_f$ be the event that that DNA test falsely identifies the person as a murderer. and $D_t$ is the event that DNA test truely identifies the murderer. So we have $$P(D_f)=\frac{1}{1000}$$ and $$P(D_t)=\frac{999}{1000}$$ Now I am really confused how to calculate the probability that the person is murderer?

Amaan M
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Umesh shankar
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  • You should use Bayes' theorem. Intuitively, the probability will be low because if a million are tested, we expect a thousand positives but only one murderer. – John Douma May 13 '21 at 18:30
  • Consider using Bayes' theorem with the events "The test is positive", "The test is negative", $M$ and $M^c$. – N. Pullbacki May 13 '21 at 18:32
  • I don't understand. The test can at most point that the person was on the scene. How can it tell whether (s)he was a murderer ? – true blue anil May 13 '21 at 18:37
  • Possibly helpful: https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker May 13 '21 at 18:42
  • The question is seriously flawed. There may be roughly $1$ murderer in $10^6$ people. But for the ongoing murder investigation, there is one specific murderer. So the probability that the person caught is the murderer depends on the population base. Even conditional probability for positive match will vary depending on the population. – Math Lover May 13 '21 at 19:52

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