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# Did the Defendant get it right?

In the Prosecution Fallacy we saw that the prosecutor wrongly confuses the probability of seeing some evidence with the probability of innocence.

The example we used was the following:

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.

We saw how this was a fallacy by arguing that out of 10 million adult males we would actually expect a  large number of people to have the matching blood type (about 10,000). However, it is also a fallacy of the defence to argue either of the following:

1. The probability Fred is innocent is  99.99% (i.e. 0.9999) because Fred is no more likely to have committed the crime than any of the other 9999 matching males; or more generally
2. That the evidence is irrelevant because it does not eliminate a large proportion of the population

The first statement is only true if there is no evidence other than the blood to link Fred to the crime. But, generally there willbe other evidence and this evidence may eliminate a significant proportion of the other 'matching' people. The second statement is false because, irrespective of other evidence, the matching blood type is 1000 times more likely if Fred is guilty than if he is innocent (and using Bayes theorem this increases the prior probability of guilt by a factor of 1000). This clearly cannot be considered irrelevant.

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

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

Two strongly recommended books that cover these issues comprehensively are:
• Evett, I. W. and Weir, B. S. (1998). Interpreting DNA evidence : statistical genetics for forensic scientists, Sinauer Associates.
• Aitken, C. G. G. and Taroni, F. (2004 ). Statistics and the evaluation of evidence for forensic scientists (2nd Edition), John Wiley & Sons, Ltd.

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Norman Fenton

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