# Find the probability that the database includes the art thief, given that a DNA match has been found

You have a database of $$25,000$$ potential criminals. The probability that this database includes the art thief is $$0.1$$. In a stroke of luck, you found a DNA sample of this thief from the crime scene. You compare this sample with a database of $$25,000$$ men. And lo and behold, a match is found! You are well aware that DNA matches are not always perfect: if you pick two different persons at random, the chance that their DNA samples would match using the current testing techniques is $$1$$ in $$10,000$$. What's the probability that the database includes the art thief, given that a DNA match has been found?

I'm sure this question uses Bayes' theorem where $$A=$$ the database includes the art thief, $$B=$$ a DNA match has been found. I need to find $$P(A|B)= P(B|A)*P(A)/P(B)$$ to calculate $$P(B)$$, there are different cases, 1). the database does include the thief, and a person matches with the DNA from the scene. 2). the database doesn't include the thief, but a person matches with the DNA.

I also want to know if the probability of two people matching is $$1/10000$$, is the probability of two people not matching $$9999/10000$$?

Given the wording of the question, I think your formulation of the problem is reasonable. However you're not really given enough information to evaluate $$\ P(B)\$$ properly. Was the sample from the crime scene merely tested against random entries in the database until a match was found, for instance, or was it tested against every entry in the database? In the latter case, the exact number of matches found is vital information which should be used to evaluate the posterior probability that the thief's DNA is in the database. Since you're not given that information, I think you can reasonably assume that something like the former procedure was used. In that case, $$\ B=$$ at least one entry in the database matches the thief's DNA.

If you also assume that when the thief's DNA is in the database it is certain to match the DNA from the crime scene, then you have $$P\big(B\,\big|A\big)=1\ .$$ On the other hand, when the thief's DNA is not in the database, each entry in the database presumably has an independent probability of $$\ \frac{1}{10000}\$$ of matching the DNA from the crime scene, and a probability of $$\ \frac{9999}{10000}\$$ of not matching it. Therefore, \begin{align} P\big(B\,\big|A^c\big)&=1-P\big(B^c\,\big|A^c\big)\\ &=1-\Big(\frac{9999}{10000}\Big)^{25000}\\ &\approx0.918\ ,\\ P(B)&=P(B\,|A)P(A)+P\big(B\,|A^c\big)P\big(A^c\big)\\ &\approx1\times0.1+0.918\times0.9\\ &=0.9262\ ,\\ P(A|B)&=\frac{P(B\,|A)P(A)}{P(B)}\\ &\approx\frac{0.1}{0.9262}\\ &\approx{0.108}\ . \end{align}

For completeness, here is the calculation for the case when the thief's DNA was tested against the whole database, and exactly $$\ n\$$ matches were found. Call this event $$\ B_n\$$.

If the thief's DNA profile is in the database, then the probability that exactly $$\ n\ge1\$$ matches will be found is just the probability that exactly $$\ n-1\$$ matches will be found with the $$\ 24999\$$ other potential criminals in the database. Thus, $$P\big(B_n\big|A\big)={24999\choose n-1}\frac{1}{10000^{n-1}}\Big(\frac{9999}{10000}\Big)^{25000-n}\ .$$ If the thief's DNA profile is not in the database, then the probability that exactly $$\ n\ge1\$$ matches will be found is just the probability that exactly $$\ n\$$ matches will be found with the $$\ 25000\$$ potential criminals in the database (none of whom is the thief). Thus $$P\big(B_n\big|A^c\big)={25000\choose n}\frac{1}{10000^n}\Big(\frac{9999}{10000}\Big)^{25000-n}\ .$$ With a little bit of elementary arithmetic, it follows that \begin{align} \frac{P\big(B_n\big|A\big)}{P\big(B_n\big|A^c\big)}&=\frac{n}{2.5}\ ,\\ \frac{P\big(B_n\big|A\big)P(A)}{P\big(B_n\big|A^c\big)P\big(A^c\big)}&=\frac{n}{9\times2.5}\\ &=\frac{n}{22.5}\ , \end{align} and hence \begin{align} P\big(A\,\big|B_n\big)&=\frac{P\big(B_n\,\big|A\big)P(A)}{P\big(B_n\big)}\\ &=\frac{P\big(B_n\,\big|A\big)P(A)}{P\big(B_n\,\big|A\big)P(A)+P\big(B_n\,\big|A^c\big)P(A^c)}\\ &=\frac{P\big(B_n\big|A\big)P(A)}{P\big(B_n\big|A^c\big)P\big(A^c\big)}\Bigg(\frac{P\big(B_n\big|A\big)P(A)}{P\big(B_n\big|A^c\big)P\big(A^c\big)}+1\Bigg)^{-1}\\ &=\frac{n}{22.5+n}\ . \end{align} For $$\ n=0\$$, $$\ P\big(B_0\,\big|\,A\big)=0\$$, and $$\ P\big(B_0\big)\ne0\$$, so $$\ P\big(A\,\big|\,B_0\big)=0\$$ also.

Answer recast to include possibility that a wrong match may be found even if thief is in the database, and to present with greater clarity and simplicity through a contingency table

If the probability of a match being wrong is $$P=1/10 000$$, that of it being correct must be $$P=9999/10 000$$

I'll also assume that

• all $$25,000$$ are lined up and tested
• exactly one match is found

Also, even if the thief is present $$(T)$$,the match might not be with the thief but with someone else, say $$X$$

Drawing up a contingency table,

$$\quad\quad\quad\quad\quad\quad T\quad|\quad T^c$$

Matches $$T\quad \;\,A \quad\quad\; C$$

Matches $$X\quad \;\,B \quad \quad\;D$$

$$A = 0.1*0.9999*0.9999^{24999}= 0.82075$$%

$$B = 0.1*.0001*24999*.0001*.9999^{24998}=0.00021$$%

$$C = 0.9*0 = 0$$%

$$D = 0.9*0.0001*25000*0.9999^{24999}= 18.46866$$%

Finally, P(thief present | match found)

$$=\dfrac{A+B}{A+B+C+D} = \dfrac{0.82075+0.00021}{0.82075+0.00021+0+18.46866}$$

$$\boxed{\approx 4.26\%}$$

• Do you mean "two different persons match" or "two same DNAs match" by "event DNA matches" Aug 7 '21 at 7:30
• If the thief is not in the database, the probability of matching will follow that of a random population Aug 7 '21 at 7:37
• is 𝑃(𝑀|𝐷) (9999/10000)^24999? is 𝑃(𝑀|𝐷𝑐) 1/10000*(9999/10000)^24999? Aug 7 '21 at 8:57