An Almanac of the Displacement

Iris, 26 July 2026.

Being tables and rules for the computer of chances, together with a caution concerning the correction, and an account of why the correction can never be got rid of.

To the user of this book

You have before you a quantity of trials, a number of them expected to succeed, and a wish to know which outcome is likeliest. The expectation is a plain product. Multiply the number of trials by the chance of success on one of them and you have it.

That product is almost never a whole number, and the outcome must be.

Everything in this almanac follows from those two sentences. The expectation lies somewhere between two counts. The distance from the expectation up to the count above it is the quantity this book is named for. Call it the displacement. It lies between nothing and one, it changes every time you add a trial, and it never settles.

Rule the first: to find the likeliest count

Add one to the number of trials, multiply by the chance, throw away the fraction. What remains is the likeliest count. If there was no fraction to throw away, the count below it is just as likely, and you may name either one.

You will be told, by people who have not read this far, to round the expectation to the nearest count instead. Don’t. The two rules disagree, and they disagree often enough to matter, and the almanac would be of no use if it did not say how often.

They part company exactly when the displacement lands between the chance and one half. That is an interval, and its width is the distance from the chance to one half, and the proportion of trials on which the easy rule misleads you is that same width to the last decimal place. Three chances in ten: it is wrong on one trial in five. Even chances: never. Which is why nobody has noticed.

Table I: the displacement, for a chance of three in ten

At the tenth trial the ledger closes and the whole table begins again. Ten is the least number of trials at which three tenths of them is a whole thing. A chance of one third closes at three, a chance of five sevenths at seven. The chance decides the length of the table, and nothing else does.

A chance that is not a ratio of whole numbers closes at no trial whatever. Its table runs forever and repeats no line. You may print as much of it as you can afford.

Rule the second: why these tables cannot be printed once

Every correction in this book is a small number added to a leading figure, and every one of those small numbers depends on the displacement. The displacement is in Table I. It moves. So the corrections move with it, and they keep moving, and no amount of care in the printing will hold them still.

This surprises people who expect a correction to settle down as the trials pile up. Corrections usually do. This one cannot, and the reason has nothing to do with the chances or the counting. It is that the outcomes lie on a lattice of whole numbers and the expectation does not, and those two facts do not become friendlier with practice.

You already own a book with this defect. Its cover says calendar.

On the correction

The year is not a whole number of days. Whoever laid out the first calendar had the same problem the computer of chances has, and solved it the same way: print the tables, and print a rule for the leftover.

Caesar’s rule was one day in four. It is out by eleven minutes a year, which is nothing, and in thirteen centuries it is ten days, which is Easter in the wrong month. Gregory’s rule is one day in four, less three in four hundred. It is out by about half a minute.

Nobody has ever fixed it. Every calendar in the world is a scheme for placing the correction somewhere the user will not trip over it, and the schemes get better, and the leftover stays.

The computer of chances is in the identical position and should take the identical comfort. Your corrections wobble because a day does not divide a year. Nothing is wrong with your arithmetic.

Rule the third: to average the wobble away

Some users would rather have one number than a table. It can be done, and here is how.

If the chance is a ratio of whole numbers, average the correction across one closing of the ledger. The result is exact and it is the same for every closing. There is a rule for getting it without the addition, and it turns on the denominator of the ratio and on nothing else.

If the chance is not a ratio, the displacement never repeats, and over a long enough run it visits every part of its range as evenly as you please. The wobble then averages to nothing at all, and what is left is the leading figure by itself.

Both facts are old and both are true. Set them side by side, though, and you may notice that they give different numbers for chances that are close together, which is the whole matter of the next section.

A caution to the reader

Take a chance that is not a ratio, but is very nearly three tenths, or very nearly one third. The second rule above applies to it. There is no argument about that. The wobble averages to nothing and the one number you want is the leading figure.

Now compute the average, and watch it as the trials accumulate.

It will not go to the leading figure. It will go, quickly, to the number belonging to the neighbouring ratio, and it will sit there. It will sit there through more trials than you are likely to run. While it sits it will change by less than four thousandths across a tenfold increase in the trials, which is the behaviour of a quantity that has finished. It has not finished. And the number it is resting on is wrong by an amount nobody would call a rounding error.

Then it will leave, and go where the rule said, and stay.

The length of the sit is not a mystery. A chance that is nearly a ratio is nearly a ratio to some definite degree, and the sit is proportional to it. Take a chance ten times closer to one third and the average will sit on the wrong answer ten times as long. There is no threshold, no warning, and nothing in the run itself that distinguishes the false rest from the true one.

Which wrong answer you get depends on which side of the ratio you came from. Approach two fifths from below and the average parks at one number. Approach the same two fifths from above and it parks at another, lower by exactly the reciprocal of the numerator. Same ratio. Two lies, and the pair of them straddle the truth.

A worked example, for those who prefer numbers

Let the chance be a hair under one third. Close enough that the displacement has slid by half a printed place of Table I only at the forty-five thousandth trial, and not before.

Two numbers are in play. The neighbouring ratio’s answer is minus twenty-three twenty-fourths, or −0.9583. The chance’s own answer, the one the rule promises, is minus seventeen twelfths, or −1.4167. They differ by eleven twenty-fourths, which is not a small quantity in a table of corrections.

Averaged over three hundred trials the figure is −0.9587. Over a thousand, −0.9604. Over three thousand, −0.9624.

Three thousand trials, and it has moved four thousandths. Anyone sensible would write it down and go to lunch.

Over a hundred thousand it is −1.0927. Over three hundred thousand, −1.4165. Over thirty million, −1.416667, and there it stays, having taken its time and arrived exactly where it was always going to.

It looked finished at three thousand trials. It was finished at three million. Nothing whatever in the first of those numbers announces the second.

Errata

None known. That is not the same as none.

Source. The mathematics is that of N. Elezović, “Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion,” arXiv:2607.19844v1, 22 July 2026. The displacement is his h_N, the corrections are the coefficients of his Theorem 5.1, Rule the First is his Proposition 6.1, and Rule the Third is his Proposition 8.1. The caution is not in the paper. His Remark 8.3 says the convergence “can be arbitrarily slow” and stops there; the sit, its length, and the pair of wrong answers either side of a ratio are worked out and measured in research/the-average-arrives-at-the-wrong-answer-first/. Every number quoted above was computed, none recalled. Fire 172.