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I have a question regarding Example $3o$ on page $78$ of $Chapter$ $3$ in Sheldon Ross's book "A first course in Probability", the problem says the following:

A crime has been committed by a solitary individual, who left some DNA at the scene of the crime. Forensic scientists who studied the recovered DNA noted that only five strands could be identified and that each innocent person, independently, would have a probability of $10−5$ of having his or her DNA match on all five strands. The district attorney supposes that the perpetrator of the crime could be any of the one million residents of the town. Ten thousand of these residents have been released from prison within the past $10$ years; consequently, a sample of their DNA is on file. Before any checking of the DNA file, the district attorney feels that each of the ten thousand ex-criminals has probability $α$ of being guilty of the new crime, while each of the remaining $990,000$ residents has probability $β$, where $α = cβ$. (That is, the district attorney supposes that each recently released convict is $c$ times as likely to be the crime’s perpetrator as is each town member who is not a recently released convict.) When the DNA that is analyzed is compared against the database of the ten thousand ex-convicts, it turns out that A. J. Jones is the only one whose DNA matches the profile. Assuming that the district attorney’s estimate of the relationship between $α$ and $β$ is accurate, what is the probability that A. J. is guilty?

Ross solves this by calculating the probablity that Jones is guilty given that he has a match in the $10,000$ ex-cons on record doing the following: $$\mathbb {P}(G | M) = \frac {\mathbb {P}(M|G) \mathbb {P}(G)}{\mathbb {P} (M)} $$ Where $G$ and $M$ are the events that he is guilty and that he is the only one who matches the DNA strands. We must expand $\mathbb {P} (M) = \mathbb {P} (M | G) \mathbb {P} (G) + \mathbb {P} (M | G^c) \mathbb {P} (G^c)$
The way he computes the last term of the equation by calculating the probability that all other patients are innocent given that AJ is innocent and then multiplying by the probability of AJ matching the DNA and no other ex-con having the DNA strands. I understand why the last part is $10^{-5}(1-10^{-5})^{9,999}$. What I do not understand is why does he multiply by the probability of all of them being innocent?
If they are all innocent, wouln't the $10^{-5}(1-10^{-5})^{9,999}$ term account for that? Why multiply by the other probability then?

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According to the book, $$ \mathbb{P}(M|G^c) = 10^{-5} \left( \frac{1-10000\alpha}{1-\alpha} \right) \left(1-10^5\right)^{9999} $$ while $\frac{1-10000\alpha}{1-\alpha}$ represents the probability of all others in the database being innocent while AJ is innocent.

Here is one approach to understand this.

We want to calculate the probability of AJ being the only match given he is innocent, which requires the following.

(a) His DNA is found on the strands, with probability $10^{-5}$

(b) The other 9999 people in the database are all innocent (because otherwise someone other than AJ would have left DNA on the strands), and none of these 9999 people's DNA is found on the strands, with probability $\displaystyle \left( \frac{1-10000\alpha}{1-\alpha} \right)\left(1-10^5\right)^{9999}$

Then we get the result for $\mathbb{P}(M|G^c)$.

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  • $\begingroup$ Yes but if all the other ex-convicts are innocent, then none of their DNAs will match, so why must we include the other term with the alphas? $\endgroup$ – Guacho Perez Apr 21 '16 at 4:31

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