a crime has
been committed and that the criminal has left some physical evidence,
such as some of their blood at the scene. Suppose the blood type is
such that only 1 in every 1000 people has the matching type. A suspect,
let's call him Fred,
who matches the blood type is put on trial. The prosecutor claims
that the probability that an innocent person has the matching blood
type is 1 in a 1000 (that's a probability of 0.001). Fred has
the matching blood type and therefore the probability that Fred is
innocent is just 1 in a 1000.
prosecutor’s assertion, which sounds convincing and could
easily sway a jury, is wrong.
To see why (jnformally) look at the animation below (if you cannot see this then try this animation (wmv
So what is the source of
and why do lawyers so commonly make it? It all boils down to a basic
misunderstanding about probability (a misunderstanding which many
intelligent people have because this kind of basic probability is never
taught at schools).
The misunderstanding is to assume that the probability of A given that
we know B is true, written P(A|B), is the same
as the probability of B given that we know A is true, written P(B|A).
(If you want an explanation of this without any maths at all, you
read this page
In this case let A be the assertion “Fred is
and let B be the assertion “Fred has the matching blood
type”. What we really
want to know is P(A|B) (the probability Fred is innocent given that he
the matching blood type) and this is what the lawyer claims
equal to 1 in a 1000. But in fact, what we actually know is that P(B|A)
(the probability Fred has the matching blood type given he is innocent)
equal to 1
in a 1000.
The lawyer has simply stated the probability P(B|A) and claimed this is
actually the probability P(A|B).
The fallacy becomes
challenging when DNA evidence is used. In such cases P(B|A) can be
extremely low, such as 1 in 10 million. When the lawyer wrongly asserts
that the probability of innocence is therefore 1 in 10 million it seems
especially convincing. But even in this case the probability of
innocence could actually be very high. Let's assume a population of
10 million people could have committed the crime. Then, assuming
Fred is one of these people, each of the other 9,999,999 has a
probability of 1 in 10 million of also having the matching DNA. By the
Binomial theorem (see here) the expected number of other people matching is about 1. So instead of the claimed 1 in
10 million probability of innocence the real probability is about 1 in 2. In such circumstances the ‘beyond
doubt’ criteria can hardly be claimed to be met.
N. E. (2011). "Science and law: Improve statistics in court." Nature
479: 36-37. Paper on Nature online website is here. An extended draft on which this was based is here.
Fenton, N.E. and Neil, M. (2011), 'Avoiding Legal Fallacies in Practice
Using Bayesian Networks', Australian Journal of Legal Philosophy 36,
114-151, 2011 ISSN 1440-4982 (extended preprint draft here).
Fenton NE and Neil M,
Observation Fallacy and the use of Bayesian Networks to present
Probabilistic Legal Arguments'', Mathematics Today ( Bulletin of the
IMA, 36(6)), 180-187, 2000 (available here)
A nice (23 minute) video of a lecture by Peter Donnelly covering this
(and other relevant issues) is here.