The “Paradox of the Gatecrasher” is not a paradox

There is a sizable scholarly literature discussing the so-called “Paradox of the Gatecrasher,” a simple thought experiment introduced many years ago by British philosopher L. Jonathan Cohen, an evidence problem designed to test the proper role of statistics in law. (For a survey of this legal literature, see footnote 8 of our latest paper “Visualizing Probabilistic Proof.”) Briefly, Wikipedia presents the gatecrasher problem as follows:

Statistical syllogisms may be used as legal evidence but it is usually believed that a legal decision should not be based solely on them. For example, in L. Jonathan Cohen‘s “gatecrasher paradox”, 499 tickets to a rodeo have been sold and 1000 people are observed in the stands. The rodeo operator sues a random attendee for non-payment of the entrance fee. The statistical syllogism:

  1. 501 of the 1000 attendees have not paid
  2. The defendant is an attendee
  3. Therefore, on the balance of probabilities, the defendant has not paid

is a sound one, but it is felt to be unjust to burden a defendant with membership of a class, without evidence that bears directly on the defendant.

Notice, however, that Professor Cohen’s Gatecrasher Paradox is not really a paradox in the true sense of the word. In fact, it’s not even a mildly interesting problem, if you are a good Bayesian, that is. Why not? Because the statistical syllogism above only establishes a Bayesian prior, i.e. the initial probability that a randomly-selected attendee snuck in the rodeo without paying for his or her ticket. We still have to update our prior! Accordingly, let’s assume that the randomly-selected rodeo attendee in the example above is a man (let’s call him Mr X) and that our hypothetical Bayesian inquisitor is a woman (let’s call her Miss B). After finding her Bayesian prior, Miss B, our Bayesian puzzle-solver, would also ask Mr X to produce some scrap of evidence or proof that he paid for his rodeo ticket — a receipt, a ticket stub, a witness, etc. — and our Bayesian sleuth would then update her prior in a manner consistent with the evidence, including the lack of any evidence if that were the case.

Let’s solve the case of the gatecrasher

This entry was posted in Bayesian Reasoning, Law, Philosophy and tagged . Bookmark the permalink.

One Response to The “Paradox of the Gatecrasher” is not a paradox

  1. Pingback: All proof is probabilistic | prior probability

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