The probability that an innocent man would match the DNA found at a crime scene is about one in a million.

" 'The question they ask of the guy in the dock is, what is the
probability that an innocent man would match this DNA?  And the answer
is, of course about one in a million', says Pike, looking around the
table and pausing for drama.  'The question they SHOULD ask is, GIVEN
THE MATCH, what is the probability that he is innocent?  And the answer
is, about one-third.' "

From Bayes theorem:
P(Guilty|Positive Test) =
          {P(Guilty) * P(Positive Test|Guilty)} / P(Positive Test)

In other words, you need the "prior" probability that he is guilty to
answer the question of his guilt or innocence, given a "match".

-- Peter M. Layde <playde@post.its.mcw.edu>

Come on, people.  No one in this discussion has yet stated the main
point, which is that the guilt or innocence of the accused is not an
event for which one can compute a probability.  He either is guilty or
he isn't.  Probability calculations need to be made in the context of a
random process.

The issue is the same as the issue in all statistical tests of
hypothesis.  There, too, one cannot ask what is the probability that
such and such an hypothesis is true (e.g., smoking does not cause lung
cancer); rather one designs the test so that if the null hypothesis is
true (e.g., smoking does not cause lung cancer), then the probability
of obtaining a result on the test as refuting of this hypothesis as it
is (e.g., lots of dead smokers compared to nonsmokers) is small (5%,
1%, whatever).

In the DNA case, either the suspect committed the crime and left his
blood at the scene, or someone else committed the crime and left his
blood at the scene.  The null hypothesis is that someone else did it.
If this is true, then that (random) person has only a 1/1000000 (or
whatever) chance of having the suspect's DNA type.  So either we have
just witnessed an amazing coincidence (the real criminal just happened
to have the suspect's DNA type), or else the suspect is the guilty
party.  Since we know that rare events don't occur very often, we feel
confident in concluding that the suspect is guilty, rather than that we
have just witnessed a very rare event.  We will never know which is the
true state of affairs, nor does it make any sense to ask for the
probability of which is true, since, as I said above, it's not a chance
event we are talking about -- either the suspect did it or he didn't.

The risk we're taking in doing this is the chance that an innocent
defendant is convicted; in other words, IF the defendant is innocent,
then we have a 1/1000000 chance of wrongly convicting him (this is
called a Type One error in statistics).  It's the risk we have to take
("proof beyond a reasonable doubt") in order to convict the guilty
ones.  Note that this calculation says nothing at all about the chance
that the particular defendant before the court is guilty, only what
happens over the long run to INNOCENT defendants.  (In our simple model
here, the chance of a Type Two error -- finding a guilty person
innocent -- is zero, since we're assuming that we always convict when
the blood matches.)

And in fact, the DNA test is even more damning, since it's not like we
picked someone off the street at random to charge with the crime -- we
picked up someone whom we suspect for other reasons is guilty.  BUT
IT'S NOT A PROBABILITY QUESTION!

-- Jerry Grossman <grossman@vela.acs.oakland.edu>

This is equivalent to the following:

You have a jar containing about a million marbles, three of which are
white, the rest black.  Of the three white marbles, two have hollow
cores; all other marbles are solid.

The probability that a randomly selected *solid* marble (parallel to an
*innocent* person) is *white* is about one in a million.

But given that you've selected a *white* marble (parallel to a DNA
match), the probability that it is *solid* (innocent) is one in three.

-- Angi Long <angilong@bach.seattleu.edu>

So, you have 2 guilty people?  (The 2 with hollow cores).  In my
opinion, the strangest thing about the DNA story is the fact that the
odds that the man is guilty is between .5 and 1.

-- Abigail <abigail@xs4all.nl>