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# Did the Prosecutor get it right?

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 (jnformally) look at the animation below (if you cannot see this then try this animation (wmv file))

The informal argument is as follows: Let's suppose that the crime could only have been committed by an adult male and that there are 100,000 such adult males. Then from this population we would actually expect about 100 to have the matching blood type. 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 99 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% (i.e. 0.99, 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. 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 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.

Chapter 13 of the Fenton and Neil book covers this material in detail. Also see:

Fenton, 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, ''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 (available here)

A nice (23 minute) video of a lecture by Peter Donnelly covering this (and other relevant issues) is here.

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Norman Fenton

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