This article in the Washington Monthly tells a startling story of a conflict between the legal system and probability. DNA tests use a certain number of genetic markers to match DNA samples. The samples are well-short of a complete sequencing, which means any randomly selected person has a 1 in a million chance of matching. When DNA testing is used on a few suspects, then the chances of matching an innocent person are pretty low. But now, governments are compiling large databases of DNA and data mining them for matches, which radically increases the odds of matching an innocent person. This would not necessarily lead to false convictions as long as juries are made aware of the consequences, but at least one judge ruled evidence that a DNA match happened by mining a database inadmissible: the jury never heard it.
(In the particular case of the article, the victim was raped and murdered, and the defendant was found in a database of sex offenders. The article claims the correct probability of a match in this case is 1 in 3. It’s not clear from the article if this is the unconditional probability of being found in a database of that size, or the conditional probability given that it was a database of sex offenders.)