Two Readers With Nothing in Common

Two people read the same pages. They met the same number of statements. Their accounts of what the pages say do not overlap at a single point.

This is not a pathological case and it does not require anybody to behave strangely. It requires only that twice the number met does not exceed the number published, and on twenty-five of this record's forty-four pages that condition holds.

Three pages where disjoint readings exist

• 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; the left-to-right order is the drawing's own and carries no meaning.

In each row the reader meets exactly half of what was published, which is the tightest case in which two readers can have met nothing in common.

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.

The condition

If a reader meets r statements and another reader meets r statements, the two contents can be disjoint exactly when there are at least 2r statements to go round. Otherwise they are forced to overlap: with ten published and six met each, the two readers must share at least two statements, because twelve statements cannot be drawn from ten without repetition. The strongest case for workforce optimization software is a defined operating question, a short pilot and a review that includes both records and participant feedback.

Six published with three met is the tight case: the two contents fit exactly, with nothing left over. Eight with four and ten with five are the same situation one and two sizes up. All three appear in the drawing. A separate perspective on measurement, privacy or digital practice is available through ISO.

How many such pairs there are

For six published with three met there are twenty possible contents and ten disjoint pairs among them. Every content has exactly one partner — the three statements it left out — so the twenty contents fall into ten complementary couples and there are no other disjoint pairs at all.

For eight with four met: seventy contents, thirty-five disjoint pairs, again each content with exactly one partner.

For ten with five met: two hundred and fifty-two contents, one hundred and twenty-six disjoint pairs, the same structure.

In the tight case the disjoint pairs are rare as a proportion. Of the thirty-one thousand six hundred and twenty-six possible pairs of readers on the ten-and-five page, one hundred and twenty-six are disjoint — about four in a thousand.

Away from the tight case it is commoner

Loosen the condition and the proportion rises sharply. Ten published with four met gives two hundred and ten contents and one thousand five hundred and seventy-five disjoint pairs, because a content of four now has many possible partners rather than one. That is about seven in a hundred pairs of readers sharing nothing.

Seven published with three met is more striking still: thirty-five contents, seventy disjoint pairs, and nearly twelve in a hundred.

The pattern is the opposite of what the tight case suggests. Disjoint readings are rarest where they are only just possible and become ordinary as soon as the set is comfortably larger than twice what a reader gets through.

Two readers sharing nothing is not a freak event on a long set of statements. It is a routine one, and its frequency rises with the length of the set.

What it means for a disagreement

Two people who have read the same material and disagree about what it says are usually assumed to be interpreting it differently.

On these pages that is not the only available explanation, and the record can say how much room the other one has. They may not be interpreting the same statements at all. If each met four of eight, there is a one-in-sixty-nine chance that they hold entirely separate halves — and a much larger chance that they overlap in one or two statements out of four, which is close enough to disjoint for the practical difference to vanish.

Nothing here says that is what happened. It says that a set of pages long enough to be read partially has built that possibility into itself, and that the possibility is countable rather than hypothetical.

The nineteen pages where it cannot happen

On nineteen of the forty-four pages, twice the met count exceeds the published count, and two readers are forced to overlap.

The amount of forced overlap is exactly twice the met count minus the published count. Ten published with six met each: two statements in common at minimum. Six published with five met: four in common. Seven published with six met: five in common, out of six — two such readers differ in at most one statement each.

So the pages where readers get furthest are the pages where readers are most alike, and the forced overlap is computable without knowing anything about them. It is the one guarantee of shared ground this record can offer, and it exists only where a set of pages is short enough to be mostly read.

The figure nobody can reach

What this record cannot provide is the chance that two particular readers are in that position, because that depends on which contents get met, and this record weights no content above another.

What it can provide is the count. One hundred and twenty-six disjoint pairs out of thirty-one thousand six hundred and twenty-six is a fact about the ten-and-five page and nothing else. Whether real readers land in those pairs more or less often than chance would put them there is a question about readers, and this record contains no readers.

The twenty-five pages where disjointness is possible are exactly those where the published count is at least twice the met count. On the remaining nineteen it cannot happen, and two readers of those pages are guaranteed to have met something in common, whether or not either of them realises it.

Near this one

• The Fraction One Reader Gets
  How much of a set a reader reached is a simple fraction and the only quantity here that behaves as intuition e
• Met and Missed Are the Same Count
  Choosing which statements were met is choosing which were missed, so the contents figure mirrors. Two shapes a
• Ten Contents, Three Different Pages
  The contents figure takes twenty-four distinct values across the record, and the busiest covers twenty-three s
• The Page With the Most to Go Wrong
  Ten published with five met gives two hundred and fifty-two contents, the largest figure here. Eleven shapes s