Limits

Nothing Here Says The Items Are Equally Likely

The second half of the coin convention, which is more often misread than the first. Counting profiles equally is not the same as believing they occur equally.

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The coin convention says that each item is a separate coin. It is routinely read as saying that each item is met about half the time, and it says nothing of the kind.

The distinction is subtle, it matters, and it is the one most likely to be lost.

Project owners can check how attendance is recorded before deciding whether added visibility is worth the administrative cost. For an independent technical reference, see Metabase.

What is actually being done

The record enumerates profiles and counts them. Each profile is one unit in the count.

That is a decision about how to count, not a claim about how often each profile occurs. Counting the two hundred and fifty-six profiles of an eight-item list one apiece is the same operation as counting the fifty-two cards in a deck one apiece — it establishes a denominator, not a frequency.

A frequency claim would be: each item is met by half of all candidates. The record never makes it, never needs it, and would have no basis for it.

Why the misreading is so easy

Because "one in two hundred and fifty-six" has the grammar of a probability. Probabilities are the familiar thing that fractions of that shape usually denote, and the mind supplies the familiar reading.

It also does not help that under one particular assumption — items met independently with probability one half each — the record's figure would be a probability. That assumption is not made here, but its availability makes the slide feel harmless.

The figure would be a probability if the world were a particular way. The record does not say the world is that way, and the figure is not a probability.

What follows from the distinction

The figure does not become wrong when the items are not equally likely. It was never about likelihood, so it has nothing to be wrong about. It remains an exact statement about the document.

The figure does become inapplicable as a prediction. Anybody wanting to know how many applicants will qualify needs something this record does not contain.

The comparison between two documents survives. Both are counted the same way, so the ordering between them is meaningful even though neither figure is a rate.

An analogy that holds

A dictionary reports that a word has eleven letters. That is exact and says nothing about how often the word is used.

Somebody who read "eleven letters" as "used eleven times per million words" would be making this record's most common misreading, and the dictionary would not be at fault.

The two statements, kept apart

They are easy to run together and worth separating explicitly.

The first: each configuration of met and unmet items is counted once. That is a counting rule, it is what fixes the denominator at two to the power of the number of items, and it is true by construction.

The second: each configuration occurs about as often as any other. That is a claim about the world, it is almost always false, and this record does not make it anywhere.

Everything on this site follows from the first and nothing from the second. The figures are counts of configurations, which is why they can be checked with a table of binomial coefficients and no knowledge of any field whatsoever.

What a weighted version would need

A probability for every configuration, which for sixteen items means sixty-five thousand five hundred and thirty-six numbers.

In practice nobody supplies those directly; they would supply a rate for each item and some description of how the items relate, and the configuration probabilities would be derived. That is a substantial modelling exercise, it is specific to one field and one moment, and it produces a figure that is better for one purpose and useless for comparison.

Unweighted counting gives up the first purpose to keep the second. Two documents can be compared, an edit can be priced, and a class of identical filters can be identified — all without a single assumption about who holds what.

Where the misreading does damage

When a figure from this record is quoted as a rate.

"This list admits one in two hundred and fifty-six" is a true statement about configurations and a false statement about applicants. Quoted the second way it sounds like a prediction, and somebody will plan around it — expecting a certain number of qualified people, or concluding that a filter must be loosened because it appears to admit almost nobody.

The figure cannot support that. It would be equally true of a list whose items nearly everybody holds and of one whose items nearly nobody does. Whenever a figure from here is passed on, the convention has to travel with it, which is why the phrase "one profile in" is used throughout rather than "one in".

Why the record is built this way anyway

Because the alternative is worse, as the page on the coin sets out: weights would have to be invented, the invention would be invisible in the output, and the result would look more authoritative while being less sound.

The record chooses the smaller, checkable claim. Its cost is precisely this page — the need to say repeatedly what the figures are not — and that cost is worth paying, because a reader who misreads a stated convention can be corrected, while a reader misled by invented weights has nothing to correct against.