# The Prosecutor’s Fallacy

## Conditional Probability in the Courtroom

#### Imagine you have been arrested for murder.

You know that you are innocent, but physical evidence at the scene of the crime matches your description. The prosecutor argues that you are guilty because the odds of finding this evidence given that you are innocent are so small that the jury should discard the probability that you did not actually commit the crime.

But those numbers don’t add up. The prosecutor has misapplied conditional probability and neglected the prior odds of you, the defendant, being guilty before they introduced the evidence.

The prosecutor’s fallacy is a courtroom misapplication of Bayes’ Theorem. Rather than ask the probability that the defendant is innocent given all the evidence, the prosecution, judge, and jury make the mistake of asking what the probability is that the evidence would occur if the defendant were innocent (a much smaller number):

P(defendant is guilty|all the evidence)

P(all the evidence|defendant is innocent)

### Bayes Theorem

To illustrate why this difference can spell life or death, imagine yourself the defendant again. You want to prove to the court that you’re really telling the truth, so you agree to a polygraph test.

Coincidentally, the same man who invented the lie detector later created Wonder Woman and her lasso of truth.

William Moulton Marston debuted his invention in the case of James Alphonso Frye, who was accused of murder in 1922.

For our simulation, we’ll take the mean of a more modern polygraph from this paper (“Accuracy estimates of the CQT range from 74% to 89% for guilty examinees, with 1% to 13% false-negatives, and 59% to 83% for innocent examinees, with a false-positive ratio varying from 10% to 23%…”)