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Genetic material is the smoking gun of the modern crime scene. Juries in criminal trials are often encouraged to think of DNA profiling as an exact science, in which telltale traces of skin, hair, and blood identify perpetrators with pinpoint accuracy and rule out any likelihood of mistaken identity. The statistics, however, tell a different story.

From the tens of thousands of genes that make up the human genome, scientists have whittled down the list to about 13 pairs that vary most widely among different people. They use these pairs to make graphs called “electropherograms,” in which each set of genes produces a peak of a certain height. The chance that two individuals could have all 13 peak pairs in common has been estimated to be about 1 in 400 trillion, many times the number of people on earth. So if two profiles are found to match perfectly at every peak, it is extremely unlikely that they don’t come from the same person, or from an identical twin.

It can happen, however, that the profile is not completely clear—for example...