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 $1000,000$ residents of the town. $10,000$ 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 $\alpha$ of being guilty of the new crime, while each of the remaining $990,000$ residents has probability $\beta$, where $\alpha$ = $c\beta$. (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 $10,000$ 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?
The above is a question from First Course in Probability by Sheldon Ross.The solved question is at page 93(of the pdf) in the linked pdf.
Now, the solution calculates $P$(all in database innocent) as $=$ $1 -10,000\alpha$ .
My Understanding : $P$(all in database innocent) = $1-$ $P$(at least one guilty in the database), but since only one can be guilty, all events ($i^{th}$ ex-criminal being guilty) are mutually exclusive, hence $P$(at least one guilty)= $\alpha+\alpha+..+\alpha$= $10000\alpha$.
Doubt : Why is $P$(all in database innocent) $\ne$ $(1 -\alpha)^{10,000}$? or Is $(1 -\alpha)^{10,000}$ $=$ $1 -10,000\alpha$?.
