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One Content, Many Impressions

Fifty notes on how many impressions one set of statements allows

What comes first

Every Position Has the Same Share

The share of arrangements whose second statement is checkable equals the share for the first. The opening is special in effect, not in frequency.

at most 3024 arrangements

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A third of the arrangements on this page open with a verifiable statement, because three of the nine published statements are verifiable.

A third of them have a verifiable statement in the second position too. And in the third. And in the fourth. A fair trial begins when the team can read more about the reporting model, understand every active setting and agree how exceptions will be discussed.

The proportion is c over n at every position in the encounter, and the count is c times P(n − 1, r − 1) at every position. Another useful point of reference for the wider question is Schema.org.

Nine published, four met, openings rising

the statementsarrangementscontents302412630241263024126
  • dashed: a statement a reader can check from outside
  • solid: published, but nothing to check it against
  • short and pale: never reached by this reader

Three shapes drawn; all of them give the same 126 different contents; the left-to-right order is the drawing's own and carries no meaning.

Three thousand and twenty-four arrangements in each row. A third of them open checkably, then two thirds, then all — and the same proportions hold for the second, third and fourth positions.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

Why the positions cannot differ

Fix a position — say the third. There are c choices for a verifiable statement to sit in it, and once it is placed, the rest of the encounter is an ordered selection from what is left, which was counted in the opening note as P(n − 1, r − 1).

That is the identical calculation as for the first position, because nothing in it referred to the first position. Position three is not distinguishable from position one by any counting argument, and the count comes out the same.

The uniformity convention is doing this work. Under it, no position is more likely to hold any particular statement than any other, which is exactly what makes the shares equal.

Then why count the opening at all

Because the positions are identical in frequency and not in effect.

The first statement a reader meets establishes what sort of set this is. It supplies the frame the rest is read inside. It is the only one read by a reader who stops after one — and the previous notes showed that such readers are the commonest kind and account for sixty-three shapes of this record.

So the opening is worth a separate count for its consequences, and the count turns out to be the same as any other position's. That is a result, not an omission: it says the publisher gets no help from structure here. There is no position that is naturally more likely to be verifiable.

The first position is not more often checkable than the fourth. It is more consequential when it is not.

What a set of pages can do about it

Only one thing, and it is not a trick of arrangement: raise c.

The share at every position is the verifiable fraction of the set. Moving statements around does not change it, because the record's convention treats every arrangement as reachable and the arithmetic has no notion of what the publisher put at the top.

This is where the convention and the practice part company most sharply. In practice a publisher absolutely can put a verifiable statement first and raise the chance a reader meets it early — for the readers who arrive at the front. For the readers who arrive at a paragraph three sections down, the publisher's first statement is not their first statement, and the convention is closer to right than it looks.

The honest summary is that the record's figure is a floor rather than a description. A set of pages that leads with something verifiable does better than c over n for some readers and exactly c over n for the rest.

The position that is special anyway

Position-independence is a fact about the counts and not about the encounter, and one position remains different for a reason outside the arithmetic.

A reader who stops after one statement has read only the first. The previous note counted sixty-three shapes of that kind, and for every one of them the opening is the whole encounter. So the first position is the only one that is sometimes the last, and a reader who abandons a set early has had an encounter the later positions never entered.

The counts cannot express that, because they count arrangements of a fixed r. What varies there is r itself, and comparing across values of r is comparing different shapes — which the record does freely in prose and never inside a single figure.

The three rows

Three verifiable of nine: one thousand and eight openings of three thousand and twenty-four, a third. Fifteen blind contents of one hundred and twenty-six.

Six verifiable: two thousand and sixteen openings, two thirds. No blind contents, because three unchecked statements cannot fill a four-statement encounter.

Nine verifiable: all three thousand and twenty-four openings, and nothing blind.

The middle row is the interesting one. It has eliminated blind encounters entirely while leaving a third of all openings unverifiable — at every position, not just the first. A reader of such a set is guaranteed to meet something auditable, and under the convention any given position of their encounter, first through fourth, holds an unverifiable statement one time in three.

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