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The problem under consideration is:

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?

Problematic Part of Solution

In order to calculate conditional probability, there is the following intermediate step

Now, let G be the event that A. J. is guilty, and let M denote the event that A. J. is the only one of the ten thousand on file to have a match.

If A. J. is guilty, then he will be the only one to have a DNA match if none of the others on file have a match. Therefore, $$P(M|G) = (1 − 10^{−5})^{9999}$$ On the other hand, if A. J. is innocent, then in order for him to be the only match, his DNA must match (which will occur with probability $10^{−5}$), all others in the database must be innocent, and none of these others can have a match.

Now, given that A. J. is innocent, the conditional probability that all the others in the database are also innocent is $$P(all\ others\ innocent\ |AJ\ innocent) = \frac{P(all\ in\ database\ innocent\ )}{ P(AJ\ innocent)}$$ $$ = \frac{1 − 10,000α}{1-\alpha}$$

Doubt

Why is ${P(all\ in\ database\ innocent\ )} = 1 − 10,000α $. Shouldn't it be $$(1-\alpha)^{10,000}$$.

Reference- Sheldon Ross, 8th Edition, Example $3o$.

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  • $\begingroup$ Quick comment... the reference is by a single author. (: Sheldon M. Ross $\endgroup$ – knrumsey Jun 9 '17 at 17:26
  • $\begingroup$ Thanks..Changed it. It just has become a habit of mine to say, Sheldon and Ross. :) $\endgroup$ – Manish Jun 9 '17 at 17:29
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I think this is what is going on: From the problem statement, it appears that we are to assume that there is one and only one perpetrator. Therefore the guilt or innocence of different individuals are not independent events. In fact the guilt of individual x is mutually exclusive of the guilt of individual y (provided x and y are different people).

Thus the probability that the guilty person is one of the released felons is $\alpha+\alpha+\cdot+\alpha = 10000\alpha$. Hence the probability that none of them is guilty is $1-10000\alpha$.

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  • $\begingroup$ So you mean to say that we should consider them as dependent but mutually exclusive cases? $\endgroup$ – Manish Jun 9 '17 at 18:01
  • $\begingroup$ Yes. The guilt of various people in the town are pairwise mutually exclusive. $\endgroup$ – paw88789 Jun 9 '17 at 18:07
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    $\begingroup$ In fact, the problem explicitly says that the crime has been committed by a solitary individual. $\endgroup$ – paw88789 Jun 9 '17 at 18:08

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