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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 States43075 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
