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I heard this fallacy but for some reason I haven't found anything that says why it's wrong. Can any1 figure it out (try not using wikipedia / google :D)?
Scenario: DNA of a suspect is found on a murder weapon. Is this enough proof to convict?
Lawyer argues: Theres a 1 in 5 million chance the DNA sample is another person's sample (match occured by chance). There are roughly 60 million people in America. This means that the tested DNA could correspond to roughly 12 other people. This means that theres a 1 in 12 chance that the suspect is guilty and not one of the other 11 people. Thats not conclusive enough to convict. I know this is a fallacy but any idea how it works?
Population number is obviously incorrect right now but might have been correct some time ago
 
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My take on it:
+ Show Spoiler +As far as the DNA test is concerned, there are two possibilities: it is right or it is wrong. That is, A) DNA belongs to suspect or B) DNA belongs to someone other than suspect.
The only relevance of the number of people who are not the suspect is to subdivide possibility B. So, chances that the DNA is the suspect: still 5 million to 1. Chances that the DNA belongs to anyone else in America? Still 1 in 5 million. Chances that the DNA belongs to any individual who is not the suspect? Around 1 in 300 billion, but no one's asking that question.
edited to add a 0, since you edited your post from 6 million to 60 million, not that the exact number matters.
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Kyrgyz Republic1462 Posts
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else.
I.e. there are other factors that suggest that the crime was likely perpetrated by the person in question, and if you took a random person with a matching DNA you could say with very high confidence that he has nothing to do with the crime due to other factors.
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pretty sure it always said 60
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On April 27 2010 04:29 paper wrote:
pretty sure it always said 60
:shrug: Then I misread it the first time.
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The lawyer would be correct if all sixty million Americans were suspects.
And the American population has not been sixty million since 1890.
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On April 27 2010 04:29 Random() wrote:
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else.
Yea, so the point is that in a country of 60 million, 12 other people have that DNA. That means it could be either of those 12 ppl.
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On April 27 2010 04:29 Random() wrote:
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else.
On April 27 2010 04:30 MoltkeWarding wrote:
The lawyer would be correct if all sixty million Americans were suspects.
I disagree with both of you. I think that the lawyer would be incorrect in any case, because the 5 million to 1 statistics measure the chances of the identification being accurate--nothing to do with whether the test was done on purpose nor whether anyone else is a suspect, nor whom the DNA might belong to if the test results were erroneous.
Edit: after reading SubtleArts second post, it seems I misunderstood the premise. It sounds like the premise is that 1 in 5 million people have identical DNA (or comparable DNA fragments, w/e), and this is what gives the test its chance of failing. In that case, I agree with Random() and Moltke.
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On April 27 2010 04:31 SubtleArt wrote:Show nested quote +On April 27 2010 04:29 Random() wrote:
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else. Yea, so the point is that in a country of 60 million, 12 other people have that DNA. That means it could be either of those 12 ppl.
Yeah but thats where other pieces of evidence come into play, no matter how small or unsubstantial simply because it helps to further eliminate any of those possibilities.
I mean, simply being in the same city where the crime took place, would technically narrow down the possibility of 12 other people by quite a bit.
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Kyrgyz Republic1462 Posts
On April 27 2010 04:31 SubtleArt wrote:Show nested quote +On April 27 2010 04:29 Random() wrote:
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else. Yea, so the point is that in a country of 60 million, 12 other people have that DNA. That means it could be either of those 12 ppl.
But only one of them was suspect. There are obviously circumstances that make those 11 others not suspect.
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On April 27 2010 04:19 SubtleArt wrote:
I heard this fallacy but for some reason I haven't found anything that says why it's wrong. Can any1 figure it out (try not using wikipedia / google :D)?
Scenario: DNA of a suspect is found on a murder weapon. Is this enough proof to convict?
Lawyer argues: Theres a 1 in 5 million chance the DNA sample is another person's sample (match occured by chance). There are roughly 60 million people in America. This means that the tested DNA could correspond to roughly 12 other people. This means that theres a 1 in 12 chance that the suspect is guilty and not one of the other 11 people. Thats not conclusive enough to convict. I know this is a fallacy but any idea how it works?
Population number is obviously incorrect right now but might have been correct some time ago
Not sure if it's just poor wording or the trick itself. If there's only a 1 in 5.000.000 chance that the sample belongs to someone else then the suspect should be convicted obviously. He almost certainly is the murder by definition.
If however you meant that the test has a 1:5.000.000 chance to give a false positive then the lawyer's argument has merit. If there's no additional evidence (i.e his name was pulled out of a complete, nationwide database as the first match) then the suspect should be acquitted. But, in this situation, every information that significantly cuts down the number of possible suspects from the initial 60 million would count as strong evidence.
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I'm no lawyer or anything like that and I suck at the terminology on this subject, but anyways:
Since a DNA test alone doesn't clearly identify a single person I wouldn't judge the case on such a test alone.
First of all it has to be clear that the DNA is the murderer's.
Secondly, an innocent person should never be found guilty, so it has to be clear that noone else could have commited that crime. A DNA test alone cannot provide 100% certainty, it should be backed up by various other proofs. Basically you have to get to a 1 : 7,000,000,000 chance.
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When it comes to deeming various substances carcinogene or not, we are satisfied with a 1:1000000 odds of a cancer developing. My point being, 1 in 5 million is already diminishingly small so with just a tiny bit of circumstantial evidence, which will always be present, judging someone largely based on DNA isn't a problem.
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Well there are 11 other people in the US who could have committed the crime and whose DNA could in fact be on the murder weapon. Considering that these 11 people could live anywhere in the US, the likelihood that the murder weapon was used by the suspect, who was in an area near the crime around the time it took place seems pretty high. When used in combination with other evidence (which created a list of suspects in the first place), the DNA test narrows down the identity of the perpetrator to one suspect. Since he is the only person who could have committed the crime, he is convicted.
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On April 27 2010 05:16 Ghostcom wrote:
When it comes to deeming various substances carcinogene or not, we are satisfied with a 1:1000000 odds of a cancer developing. My point being, 1 in 5 million is already diminishingly small so with just a tiny bit of circumstantial evidence, which will always be present, judging someone largely based on DNA isn't a problem.
That is absolutely not how carcinogens are found.
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This argument was used by OJ simpsons lawyers wasn't it? Yes if there is ONLY DNA evidence then there is an equal chance of any of the 12 or however many other people the DNA can match. However, if you have even a single additional type of evidence pointing to one person, then the argument against that person has suddenly become more or less insurmountable.
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Wait...since when DNA of a person found on murder weapon = this person is the murderer?
If somebody killed me with my brother's knife, there probably would be a hell of his DNA on it.
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On April 27 2010 04:31 SubtleArt wrote:Show nested quote +On April 27 2010 04:29 Random() wrote:
The DNA found on the weapon was purposefully compared to the DNA of one particular person. This means there is only 1 in 5 million chance that the DNA belongs to someone else. Yea, so the point is that in a country of 60 million, 12 other people have that DNA. That means it could be either of those 12 ppl.
but none of those other 12 people are being judged... you got lead to that one for a reason... and the chances of finding the othersa are 12 / 300 million
Edit: was watching south park and took over 1 hour to reply (got beat to it)
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It's stating that his interpretation of the statistics is incorrect and he's judging the probability that the suspect commit the crime improperly. Just because statistics say something, doesn't mean that the suspect has a 1 in 12 chance.
It basically oversimplifies statistics, and doesn't take everything into account. What was the probability that the man would even touch the gun? What was the probability that the 1 in 12 would even come in contact with the gun? What about taking into account that there could be more matches?
It's an incomplete statistical analysis, and is weak to be used as a basis.
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On April 27 2010 05:26 seppolevne wrote:Show nested quote +On April 27 2010 05:16 Ghostcom wrote:
When it comes to deeming various substances carcinogene or not, we are satisfied with a 1:1000000 odds of a cancer developing. My point being, 1 in 5 million is already diminishingly small so with just a tiny bit of circumstantial evidence, which will always be present, judging someone largely based on DNA isn't a problem. That is absolutely not how carcinogens are found.
Not to go completely OT, but yes the definition of when something is not carcinogene is if the incidence of cancer is less than 1 out of 1000000, simply because the incidence in that case is so small that we are having trouble isolating it from other factors.
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I feel like there could be a number of things wrong with the reasoning.
First of all, location. In a single city with a population even the size of New York's, what are the chances of a suspect matching the DNA of another person in New York? I feel like comparing that 1 in 5 million with the rest of the U.S. (60 million.. that must have been a long time ago) would not be a correct comparison in this aspect unless the circumstances behind the case were extreme.
Second, we're talking about a one in five million chance, but don't you have to consider the other people. Not just location, but age, situation, conditions, etc.
And then along the reasons that some of the other posters have decreed as well. I'm not lawyer, but that's my take on the fallacy.
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I'd buy it. He's innocent in my book.
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United States3824 Posts
Hot_Bid, why don't you give out free legal advice? 
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The statement uses words to trick you into believing faulty probability. The expected number of people that match the test is 12, but that doesn't mean that the one suspect has only a 1/12 chance of being the actual match. The key here is that the DNA could only have come from some small fraction of the population.
Disclaimer: I have not quite verified the following calculations
+ Show Spoiler [Math] +
Defining Terms:
A = DNA of suspect
B = DNA in test sample
X = test result of suspect's DNA
Y = test result of sample DNA
Let's parse the statement for raw probabilities.
There is a 1 in 5 million chances of a match despite not having matching DNA:
P((X=Y) | (A != B)) = 1/5,000,000
There are 60 million people in the US. From this we generously infer that the probability of the test sample being from a given individual is 1 in 60 million.
P(A = B) = 1/60,000,000
P(A != B) = 59,999,999/60,000,000
Also, we will assume there is no possibility of a a false negative.
P((X=Y) | (A=B)) = 1
Now for the solving. What we want to figure out is the chance that the suspect's DNA does match the test DNA given that the test is positive for a match.
The Goal: P((A = B) | (X = Y))
P((A=B) | (X=Y)) = P((A=B) & (X=Y)) / (P(X=Y)
P((A=B) | (X=Y)) = P((X=Y) | (A=B)) * P(A=B) / P(X=Y)
P((A=B) | (X=Y)) = 1 * (1/60000000) / P(X=Y)
Now to solve for P(X=Y):
P(X=Y) = P((A=B) & (X=Y)) + P((A != B) & (X=Y))
P(X=Y) = P((A=B) & (X=Y)) + P((X=Y) | (A != B)) * P(A != B)
P(X=Y) = (1/60000000) + (1/5000000) * (59,000,000/60,000,000)
P(X=Y) = 2.167 * 10^-7
P((A=B) | (X=Y)) = (1/60000000) / P(X=Y)
P((A=B) | (X=Y)) = .0769
Wow 7.69% accuracy, that's not very good at all.
HOWEVER, this was with the assumption that the blood on the gun could have come from anyone in the US. In reality, there were probably only a handful of people who ever touched the gun, which greatly changes the P(A=B).
If we assume only 30000 people have ever touched the gun, these would be the new assumptions:
P(A = B) = 1/30000
P(A != B) = 29999/30000
Using these new numbers, our result is:
P((A=B) | (X=Y)) = .995
99.5% confidence is pretty good, and that's with a still conservative estimate of 30000 possible sources for the DNA sample.
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To completely verify this, you would have to take into account every single person that has a solid alibi to prove they aren't even a suspect, every person who was far enough away that they couldn't possibly have been suspects, etc. etc.
If you ignore motive completely (i.e. anyone within the range is a suspect), then it probably drops it to maybe 1 in 10,000. But definitely not down to 1 in 12.
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It's funny to see all the people doing math to prove that the odds are probably closer to 1 in a billion than 1 in 5 million as if it really matters. It's guilty beyond a reasonable doubt, it's not guilty beyond a shadow of a doubt. 1 in 5 million is just as guilty as 1 in a billion in the eyes of an average jury.
edit: actually I didn't read OP far enough to see what he was asking  It's hard to believe he can't figure that out on his own. It's like finding a half-eaten corpse in Jeffrey Dahmer's freezer with his DNA all over the bite marks and making the argument, "well there is some guy in hawaii with the same DNA as Jeffrey Dahmer so it's just as likely that he is the murderer" Cmon guy
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Doesnt everyone have their own DNA? And from what I remember from high school no 2 people have the same DNA.
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60 million people in usa?
try 300 million?
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On April 27 2010 07:34 fulmetljaket wrote:
60 million people in usa?
try 300 million?
lol, some people just have to be internet intellectuals. Hypothetical situation, brother.
The problem is there are other reasons the 11 are not considered.
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United States42180 Posts
On April 27 2010 08:56 Romantic wrote:Show nested quote +On April 27 2010 07:34 fulmetljaket wrote:
60 million people in usa?
try 300 million? lol, some people just have to be internet intellectuals. Hypothetical situation, brother. The problem is there are other reasons the 11 are not considered.
You can only say this if you know who the other 11 are. Only if you DNA test everyone, find every guy who matches and rule out all but your suspect does that make a difference. The 12 could all be in the same town and all have legitimate reasons for handling potential murder weapons. It's statistically unlikely but if you don't know who the other 11 are then you can't dismiss them.
I'm fine with DNA being used to confirm a suspect after you already have motive + witnesses but if you use it as the starting point, such as when someone gives DNA for an unrelated offence and gets flagged on a registry, you will eventually get false positives.
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HOWEVER, this was with the assumption that the blood on the gun could have come from anyone in the US. In reality, there were probably only a handful of people who ever touched the gun, which greatly changes the P(A=B).
The number of people who actually touched the gun doesn't matter. What matters is the a priori probability distribution of suspicion over the whole population. Usually it's enough to know the probability of the suspect being the actual killer before the DNA test. If it's on the order of 1/100.000 then the evidence is probably good enough to convict. If it's around 1/1.000.000 or even lower then it isn't.
The problem is that humans are generally not very good at distinguishing between two very low probabilities. Actually many couldn't even clearly understand the difference between a priori and a posteriori probabilities. The chances of a jury getting this right isn't much better than 50% in my opinion.
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EPT
