What is the Probability of a DNA Match in Crime Scene Forensics?

[Dooga Blackrazor]

What is the value of prob(conclusion C1 is true IF premise P1 is true)? It depends on the prior probability that the suspect's DNA would be found at the crime scene -- which makes it a true match for some crime scene DNA -- and the true and false positive rates for the police forsensics lab reports for DNA matches. Suppose the prior probability that the suspect's DNA would be found at the crime scene is 0.10.* Suppose the true positive rate for the police forensics lab reports for DNA matches is 0.999. Finally suppose that the false positive rate is 0.005.**

Enter answer as number in box below. Number should be rounded to two decimals (eg 1.00, 0.99, 0.50, 0.00)

*We are sure of ten suspects based on evidence other than the DNA match. There is no more evidence for one than the other. The crime is one where the suspect would inevitably leave something containing his or her DNA.
**We have information or facts from testing about the false positive rate for some labs. This includes the lab that did DNA matching in the O J Simpson murder case. These numbers are representative. We do not have information about the true positive rate. These numbers are guesses.

P1 IS: The forensic lab says "This suspect's DNA matches some of the crime scene DNA"

C1 IS: The suspect's DNA matches some of the crime seen DNA.

The answer is 0.96 or 96%. Normally I can work backwards to find out how the answer is found, but I am totally confused and lost on this one. Been going at it for an hour. This is from a practice quiz, btw, so you are only helping me prepare for a future quiz - not giving me assignment marks or anything.

Thanks

 

[Hurkyl]

Can you write down, symbolically, everything the problem tells you?

 

[Dooga Blackrazor]

Can you write down, symbolically, everything the problem tells you?

I wouldn't know how to write them down symbolically. As far as I can tell, I could write this:

True positive rate: 0.999
False positive rate: 0.005
Probability DNA would be found at crime scene: 0.10
# Suspects: 10

Seeking to find the probability DNA will match some of the DNA at the crime scene.

Now I assume this means that DNA was found out the crime seen and the 0.10 is no longer a variable, but I am lost really and unsure about that, too.

 

[Dooga Blackrazor]

Wow. I am such a newb. I just read the "This Forum is not for Homework Questions." Technically, it isn't homework, but if it is more appropriate to move it please do so. By the way, I also have msn at dooga16@hotmail.com if anyone finds it easier to help me with it then. Basically, if someone could put the variables into the Baye's formula or solve it with all the work shown, I could easily figure out how to do identical problems. Anything, though, is helpful.

 

[Hurkyl]

I wouldn't know how to write them down symbolically.

Well, one is obvious

P(The crime scene DNA is the suspects DNA) = 1/10.

True positive rate: 0.999
False positive rate: 0.005

So, what are true and false positive rates?

Basically, if someone could put the variables into the Baye's formula or solve it with all the work shown, I could easily figure out how to do identical problems.

(Based upon what I've seen so far in this thread) I really don't think that's where your problem lies; it looks like your problem is in converting the word problem into an algebra problem... not with the solution of the algebra problem!

 

[Dooga Blackrazor]

Is the 10% given in the question the same as the 1/10 figure you are coming up with or a different one? I am inclined to think different though I have been up awhile

True positive rate: 0.999 (gives a positive when it is positive)
False positive rate: 0.005 (gives a positive when it is negative)

Gives a positive when it is negative (0.001) False positive
Gives a negative when it is negative (0.995) True negative

Is that what you mean or something else? Or did you want it in percent? Like 99% and .5%?

 

[Dooga Blackrazor]

Baye's Theorem Notes:

P(H1|E) = P(H1|E) x P(H1) divided by P(H1|E) x P(H1) + P(E|H2) x P(H2)

probability Match / Lab Match =

prob(lab match/probmatch) x probmatch

_________________ (division)

prob(lab match/probmatch) x probmatch + prob(labmatch/nomatch) x prob(nomatch)

Trying to put these into terms I can understand better:

The probability there is a match (in reality) and a match (according to test) is what I am solving for. I will call this X.

X = true positive rate x probability there is a match divided by itself + false positive rate x probability there is no match.

So, now I am trying to find out what the probability of there being a match or no match is. There, I am a bit lost. I will go with 1/10 and 9/10th to see how that works for me.

Well, that is troubling. I seem to have the answer. If I did it incorrectly and got the right answer, that would be helpful to know as well.

Thanks for the help.

 

[Hurkyl]

Is the 10% given in the question the same as the 1/10 figure you are coming up with or a different one?

Same one.

Is that what you mean or something else?

Something else. For example, I want you to write down what it means for the false positive rate to be 0.005 in the form:

The probability of ________ is ____​

or maybe

The probability of _______ given that __________ is _____​

(or, preferably, the equivalent statement using P notation)

 

[Dooga Blackrazor]

Well, I got it and was able to answer all the questions of that kind on a quiz. Thanks