Suppose 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.

But the prosecutor’s assertion, which sounds convincing and could easily sway a jury, is wrong.

To see why, let's suppose that the crime could only have been committed by an adult male and that in the population there are 10 million adult males. Then from this population we would actually expect a large number of people who have the matching blood type (about 10,000). If there is no evidence other than the blood to link Fred to the crime then Fred is no more likely than any of the other 9999 matching blood type men to have committed the crime (but please see also the defendent's fallacy). This means that the probability Fred is innocent is actually 99.99% (i.e. 0.9999, which is rather different from the 0.001 claimed by the prosecution).

So what is the source of the fallacy 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 should read this page first)

In this case let A be the assertion “Fred is innocent” 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 has the matching blood type) and this is what the lawyer claims is 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) is 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 especially 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. Assuming again a population of 10 million people who could have committed the crime. Then it turns out that the probability of more than one person having the matching DNA is still actually quite high - about 0.46. So instead of the claimed 1 in 10 million probability of innocence the real probability is not much less than 1 in 2. In such circumstances the ‘beyond reasonable doubt’ criteria can hardly be claimed to be met.

See the page on the fallacy of reasoning about evidence in Court for further information about all of this.

For more information on fallacies in legal reasoning read this article:

Fenton NE and Neil M, ''The Jury Observation Fallacy and the use of Bayesian Networks to present Probabilistic Legal Arguments'', Mathematics Today ( Bulletin of the IMA, 36(6)), 180-187, 2000.

which is available here:

http://www.agenarisk.com/resources/white_papers/jury_fallacy_revised.pdf