“We the Jury ...”
(Conditional Probability in Criminal Justice)

Conditional probability can be literally a matter of life or death.

For example, suppose you are on the jury in a murder trial. The prosecution introduces DNA evidence from the scene, and establishes with expert testimony that “the odds are a billion to 1 against anyone else sharing this DNA.”

You might think that means the odds are a billion to 1 against the defendant, or the probability is a negligible 1/1,000,000,000 that he’s innocent. But actually that’s the probability that he shares this DNA if (given that) he’s innocent, not the probability that he’s innocent if he shares this DNA.

Mixing up these two is known as the Prosecutor’s Fallacy. Sometimes out of ignorance, sometimes out of a misplaced desire to win, prosecturs often introduce a correct probabiliy that something is very unlikely if he defendant is innocent, but get the jury to believe that the defendant is very unlikely to be innocent if that thing happens.

In fact, if the odds are a billion to 1 against sharing this DNA, then there are likely to be five other people out of the world’s population of 6 billion who do share this DNA. Since five of the six are certainly innocent, the defendant’s odds of being innocent if he shares this DNA are 5 in 6, not 1 in a billion.

You might object, “But those other five people could be on the other side of the planet when the murder happened!” That’s a good thing to wonder about, and the defense would certainly have to address that. In the meantime, here’s an example that doesn’t have that problem, from page 76 of John Allen Paulos’s book Once upon a Number:

Imagine a city of approximately one million people, a heinous murder has been committed, and the only evidence available indicates that the murderer has a very rare sort of mustache. [You could pick any other physical characteristic if “mustache” seems far fetched to you — a distinctive scar, perhaps, or extreme height. —SB]

Assume further that only two residents of the city have such mustaches. One of these people is innocent, the other guilty. Then the probability that an innocent person has this rare form of mustache is one in a million; one out of the one million people has such a mustache. By contrast, the probability that a person having such a mustache is innocent is one in two! [Two people have such mustaches, one of them is innocent.] Circumstantial evidence, motive, and further physical evidence therefore should always be sought to bolster any single piece of forensic evidence.

Remember that when you’re on a jury!