The jury is convinced. Despite the fact that the suspect continues to plea not guilty (but hey, that's to be expected), a guilty verdict is returned. The jurors' motivation is: "The defendant simply must be the one who did it. Why, there machine is almost error-safe! The chance of a bad decision are only one in a million. That's good enough for the court, We, the jury, are convicting the defendant."

The truth here is that the suspect probably didn't do it. If the fingerprint maching machine makes an error once in a million comparisons, then statistically it would spit out 10 suspects, since there are 10 million citizens. The chance that any randomly chosen suspect who is identified by the machine has committed the crime, is therefore 10%. The chance that the John Doe who was arrested and convicted is actually not guilty, is 90%.

The error that was made by the jurors is a typical one. The "one in a million" error chance suggests that the fingerprinting method is pretty accurate. Which it may or may not be - but in any case, that's not the question here. Bayes described his theorem using the following phrasing: "What are the odds that X happens, given the fact that Y is already observed". The "given that" clause is paramount: the jurors should have asked themselves, "what are the odds that the defendant is not guilty given the fact that his fingerprints matched". The answer to this question is straight forward: we expect 10 people's fingerprints to match, and only one of them is the perp - so there's a 90% chance that the defendant is not guilty. The given-that clause will typically decrease the size of the reference population by selecting out a group with an already known factor.

I always try to illustrate this for myself as a matrix of categories and (sub)populations. We have two factors here: (a) A person is guilty or not, and (b) a person's fingerprints match or they don't. These categories can be illustrated in a simple 2x2 matrix, with a third row and column for totals:

I think that this representation nicely summarizes all of the above, though admittedly it takes some time to getting used to such illustrations. The bottom line of this example is that large data collections (such as a database of fingerprints of all citizens) may not be useful at all in determining culprits when used as the sole source of information. On the contrary: "very precise" methods and large datasets may look good but produce very bad results. Every method, however precise, has a chance of producing false positives (see also the Schiphol entry), and even a small chance of false positives will yield quite large numbers of distinct cases when the reference population is large enough.

As a last remark. Incase you're wondering how precise fingerprint comparisons are: current methods are approx. 98.5% accurate or better, which is way less than the hypothetical 99.9999% used in this example. That's another reason why I think that Japan's plans to fingerprint all foreigners make no sense.