Going round

A Share Is Not a Chance

Every figure in this column counts outcomes, not events. Turning one into a probability needs an assumption the record declares and does not make. Closing this part.

2562560 outcomes

Home · Going round · Caution

Nine hundred and seventy-one thousand four hundred outcomes of two million five hundred and sixty-two thousand five hundred and sixty contain exactly one circular triple. That is about three in eight.

No set of comparisons therefore has a three-in-eight chance of holding one. The figure counts outcomes, and this note is about what separates those two statements. A team translating this comparison into a limited operational pilot can look at the implementation notes before deciding which time and project records are genuinely necessary.

Six candidates, nine pairs compared

column height: how many outcomes have exactly that many circular triples
the figure under each: how many triples came out circular

Six candidates with nine pairs compared. The columns drawn are every one the page has, and add to all 2 562 560 of its outcomes. 838 130 of the 2 562 560 fall into an order. The shortest column is about 171 times shorter than the tallest, and the scale is linear. For an independent perspective on comparison, evidence or decision design, consult Microsoft Power BI.

Five columns, one for each possible number of circular triples. They add to all two million five hundred and sixty-two thousand five hundred and sixty outcomes the page allows.

The column is the count of outcomes with exactly that many circular triples. The columns of one page add to the page's own total, and the drawing checks it.

What the figure counts

Arrangements, and nothing else. Among all the ways nine comparisons on six candidates could be picked and could land, this many carry exactly one circle.

Turning that into a likelihood takes one more move — supposing that every one of those arrangements is as reachable as every other. That supposition is the third of this record's declared conventions. It goes undefended because nothing it describes exists.

How real comparing differs

It is not uniform and it is not random.

Somebody comparing candidates picks the pairs for reasons. They often compare the two that seem closest, or the ones they have information about, or whichever two are in front of them. They have seen the earlier results when they make the later comparisons, and what they have seen shapes what they look for.

Arrangements that nobody would ever produce are counted here exactly the same as the ones that come up constantly.

Which way the error runs

Toward fewer circles, almost certainly, and the record cannot say by how much.

A comparer who has decided the first candidate is the strongest tends to see them win the next comparison too. That consistency is partly real and partly expectation, and either way it pushes the results toward the clean arrangements, which are a small corner of the space counted here.

So the shares in this part probably overstate how often circles occur in practice. Probably, by an unknown amount, in a direction that depends on who is comparing — which is three qualifications too many for the figure to be used as a rate.

A share measures the room. Which part of the room a particular set of comparisons ends up in is a fact about the person who made them, and nobody here has met that person.

What a procedure matching the convention would look like

It is worth knowing that one exists, because it shows how strange the convention is.

Pick nine of the fifteen pairs at random, every set of nine equally likely. Compare each of the nine by tossing a coin. Record the result.

Do that and the shares printed on this page are the exact frequencies: about three in eight of the runs will contain exactly one circular triple, and over enough runs the proportion will settle there.

Now notice what the procedure involves. The pairs are chosen without looking at the candidates. Each comparison is decided without reference to any other. Nothing is revisited when the results look odd.

Nothing anybody does resembles that, and nobody would call it comparing. So the uniformity convention does not claim that real comparing comes close to it. It claims only that the counts are exact for a procedure nobody performs.

Why no probability is printed

Weights could be invented and a rate published by the afternoon. The result would read more smoothly and mean less.

Every figure after it would then hold two things at once: the arithmetic, and a guess about how people compare. The printed number would show no seam between them, and a reader who doubted the guess could not recompute without it.

Bare counts fail the other way round, which is the better failure: the error leans in a known direction, its size is unknown, and the printed convention puts both of those where a reader can see them.

What survives the objection

Three things in this part, and they are the ones worth carrying.

Below three comparisons a circle cannot occur, whatever the pool and however the comparing is done. That is a comparison of two integers.

A circle needs three comparisons on one triple, so the number of triples with all three pairs compared bounds how many circles there can be. That is a fact about the shape of the comparing, not about the comparer.

And the ceiling: six candidates fully compared can hold eight circles and no more, however the results fall.

The sentence this note exists to prevent

"In three sets of comparisons out of eight, there is a contradiction."

Will be. Is. Both turn a count of arrangements into a claim about events, which nothing here supports. What can be defended: among the two million five hundred and sixty-two thousand five hundred and sixty ways nine comparisons on six candidates can be picked and land, nine hundred and seventy-one thousand four hundred hold exactly one circular triple — and which of them any particular exercise produces turns on how it was carried out.

Longer, duller, and the only version that stands up to being checked.