The Prosecutor's Fallacy: How Probability Gets Weaponized in Courtrooms (and Boardrooms)

Sally Clark spent three years in a British prison for murdering her two infant sons. She didn't murder them. What killed her children, and nearly destroyed her life, was a probability calculation presented with complete confidence by an expert witness who got the math catastrophically wrong.

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This is the prosecutor's fallacy. And it shows up far beyond courtrooms.

The Error, Precisely Stated

Here's the sleight of hand: confusing the probability of the evidence given innocence with the probability of innocence given the evidence.

These are not the same thing. Not even close.

In symbols:

• P(evidence | innocent) ≠ P(innocent | evidence)

In Clark's case, pediatrician Roy Meadow testified that the probability of two children in the same family dying of SIDS was roughly 1 in 73 million. The jury heard that number and drew the obvious conclusion, someone this unlikely to be innocent must be guilty.

Two problems. First, Meadow multiplied two probabilities together as if the deaths were independent events, ignoring that SIDS risk factors run in families. The 1-in-73-million figure was already wrong before the fallacy even kicked in.

Then came the fallacy itself: even if you accept that number, it tells you the probability of observing two SIDS deaths assuming innocence. It says nothing, literally nothing, about the probability that Clark was guilty. To get that, you'd need to weigh the rarity of two SIDS deaths against the rarity of a mother murdering both her children. Which is also rare. Extremely rare. The fallacy skips that comparison entirely.

Why This Breaks Our Brains

Conditional probability runs backwards from how we naturally think. You start with what you observed and want to reason toward the cause. But probability statements are typically constructed the other way, from assumed cause toward expected observation.

Flipping that direction requires Bayes' theorem. Most people, including expert witnesses apparently, do not flip it.

graph TD
    A[Observed Evidence] --> B{What are we actually asking?}
    B --> C[P(evidence | innocent), what the test gives you]
    B --> D[P(innocent | evidence), what you actually need]
    C --> E[These are NOT interchangeable]
    D --> E
    E --> F[You need base rates to connect them]

The diagram above isn't complicated. But under pressure, in a courtroom, in a quarterly business review, in a medical diagnosis, people collapse C and D into the same question. They aren't.

It's Not Just a Legal Problem

Suppose your spam filter flags an email as spam. The filter is 99% accurate. You think: there's a 99% chance this email is spam.

Wrong. If only 1 in 1,000 emails you receive is actually spam, a 99% accurate filter still produces mostly false positives. Run the numbers: out of 1,000 emails, roughly 1 is real spam (correctly flagged) and about 10 legitimate emails get flagged incorrectly. When the filter fires, the probability the email is genuinely spam is closer to 9%, not 99%.

Same structure. Same error. Different stakes.

Data teams do this constantly with anomaly detection, fraud scoring, and model outputs. A model flags a transaction as fraudulent with 95% confidence. Someone treats that as meaning there's a 95% chance of fraud. But if fraud is rare, say, 0.1% of transactions, even a highly accurate model will generate a flood of false positives, and most flagged transactions will be clean.

The Fix Is Uncomfortable

You have to care about base rates. That's it. That's the whole fix, and people resist it because base rates feel abstract while evidence feels concrete.

When you see a test result, a model score, or a statistical finding, ask: what was the prior probability before this evidence arrived? Then ask whether the evidence actually shifts that prior enough to matter.

If the base rate of fraud is 0.1%, your model needs to be extraordinarily precise before its flags become trustworthy. If two-infant SIDS deaths are rare but double infant homicide is also rare, you cannot simply declare guilt by citing one improbable number.

Sally Clark was eventually acquitted, after two years in prison, after an appeal that required actual statisticians to explain what went wrong. She died four years later. The false conviction almost certainly contributed.

A number presented with authority feels like a verdict. It isn't. Probability is a ratio between two things, and if you only measure one of them, you're not doing statistics, you're doing theater.

The mean is lying to you. So, sometimes, is the expert on the stand.