Forensic Evidence

Probability · Axiom Academy

DNA matches, the prosecutor's fallacy, and why conditional probability matters in court A prosecutor stands before a jury and presents DNA evidence found at a crime scene: "Ladies and gentlemen, the DNA found at the crime scene matches the defendant's DNA. The probability of this match occurring by random chance is only 1 in 1,000,000 . This means there is only a 0.0001% chance the defendant is innocent!" This sounds incredibly convincing. A one-in-a-million match seems like proof beyond reasonable doubt. Is the prosecutor's reasoning correct? Does a 1-in-1,000,000 DNA match really mean there's only a 0.0001% chance of innocence? The prosecutor has made a critical error - one that has led to wrongful convictions. This mistake is called the Prosecutor's Fallacy . The prosecutor confused P(match | innocent) with P(innocent | match) . These are NOT the same thing! Probability of match IF innocent = 1 in 1,000,000 Probability of innocence GIVEN the match = ??? These are opposite conditional probabilities! Just because P(match | innocent) is small doesn't mean P(innocent | match) is small. Why It Matters: The Population To find P(innocent | match), we need to consider the entire population that could have left the DNA. Scenario: Crime in a city of 1,000,000 people Now let's apply the DNA test... Even with a 1-in-1,000,000 match, there's approximately a 50% chance the defendant is innocent ! Not 0.0001% as the prosecutor claimed. The Correct Math: Bayes' Theorem

This is the written version of the interactive lesson above. See the full Probability course.