The Chord

Draw a circle, and inside it the largest equilateral triangle it will hold, each corner touching the rim. Now draw a chord — any straight line from one point of the circle to another — at random. What is the chance it comes out longer than a side of the triangle?

It sounds like a question with an answer. There is a circle, fixed; a triangle, fixed; a clean comparison, longer or shorter; and the soft, familiar hedge "at random," which we use a thousand times a day and never think to inspect. Ask it of a mathematician and you will get an answer, confidently. Ask it of three, and you may get three.

Whichever way you draw it, a chord clears the triangle's side exactly when it passes nearer the center than the side itself runs — and the side runs at half the radius from the center. So the problem is really a single question: how often does a chord chosen at random pass within half a radius of the center? Watch three careful people answer it, each choosing at random, each without error.

The first drops two points independently on the rim, anywhere, with no preference, and joins them. Fix one of them at a corner of the triangle — you may, since nothing distinguishes one point of the rim from another — and the chord beats the side exactly when the second point falls on the far arc, the third of the rim lying between the other two corners. A third of the circle, a third of the time. One in three.

The second remembers that every chord has a midpoint, and the midpoint lies somewhere along some radius. So he picks a radius, pointing any direction, and then a point along it, uniformly from center to rim, to serve as the midpoint; the chord crosses square through it. That chord beats the side when its midpoint falls nearer the center than half the radius — and the midpoint was spread evenly from center to rim, so it lands in the near half exactly half the time. One in two.

The third also works through the midpoint, but reasons that the midpoint is a point loose in the disk, so he scatters it uniformly across the disk's area. Now the chord beats the side when the midpoint lands inside the smaller circle of half the radius, drawn around the same center. But area grows as the square of the radius, so a circle of half the radius encloses only a quarter of the disk. One in four.

Notice what the second and third have done: they asked the identical question — is the midpoint within half a radius of the center? — and got one in two and one in four. Nothing separates their answers but the sense in which the midpoint was "spread evenly": evenly along a line, or evenly across a plane. The same event, weighed two ways, has two probabilities.

One in three, one in two, one in four. The same circle, the same triangle, the same sentence — and three numbers, each derived without error, each defensible, each true.

The temptation is to hunt for the mistake, to decide that two of the three have fooled themselves and one is right. There is no mistake, and there is no right one. What differs is not the arithmetic but the meaning quietly assigned to four small words. Spread the endpoints evenly and you get one world; spread the midpoint evenly along a radius, another; spread it evenly across the disk, a third. Each is a perfectly good sense of "at random," and the phrase does not choose among them. It sounds like a single instruction. It is a family of instructions wearing one coat, and you do not notice which one you have put on until your answer comes out a different size than your neighbor's.

This was exactly Joseph Bertrand's point when he set the problem down in his 1889 treatise on probability. He was not posing a riddle to be cracked; he was filing an objection. The comfortable rule for handling ignorance — when you have no reason to favor one case over another, weigh them equally — works cleanly when the cases are few and countable, a coin's two faces, a die's six. Let the cases become a continuum, every chord in a circle, and the rule quietly comes apart, because "weigh them equally" no longer says what to weigh. Equally over the endpoints, or over the midpoints, or over the area? These are not the same thing, and nothing in the words tells you which. The rule has a name — the principle of indifference, as Keynes later called Laplace's older principle of insufficient reason — and the exact shape of its failure is worth seeing: uniform along a radius and uniform across an area are both perfectly even-handed, but the map from the one quantity to the other bends, so that flatness in the first becomes a slope in the second. To be indifferent, it turns out, you must first say indifferent about what — and that choice is the whole content of the answer.

The most beautiful attempt to rescue a single answer belongs to the physicist E. T. Jaynes, in a 1973 paper with the pointed title "The Well-Posed Problem." His move was to take seriously what the question withholds. It never says how big the circle is, or where it sits, or how it is turned. So an answer that deserves to be called objective, he argued, should not depend on the circle's size or place or angle either — it should survive shrinking the circle, sliding it across the table, spinning it. And this is not a small thing to have noticed, because those three freedoms — scale, position, rotation — are precisely the three the problem declined to pin down. Jaynes showed that exactly one way of choosing a random chord is blind to all three, and it is the second man's, the even radius, the one in two. He even scattered straws across a circle drawn on the floor and found they fell as his distribution predicted. Of the three answers, his has the best claim to being the natural one; where the problem has to pick a number, this is the number the field tends to reach for.

And still it does not close the question — for a reason more stubborn than any rival calculation. Grant Jaynes everything: that scale, position, and rotation are the obvious symmetries, that his is the most natural answer of the three. It remains true that "the answer must not depend on the circle's size" is a requirement he brought to the sentence, not one he found inside it. The bare problem asks for a random chord; it says nothing, one way or the other, about whether the recipe should care about size. That the added premise is reasonable — beautiful, even — does not make it given, and a differently chosen symmetry yields a different chord and a different number. Jaynes did not discover the meaning hiding in the word "random"; he narrowed the word by supplying a symmetry. He proved, more cleanly than Bertrand had managed, the very thing the paradox was about: that you cannot get an answer out of the question without first putting one in.

And that is the whole of it, and it is larger than a circle. "At random" belongs to a family of words that feel like finished instructions and are not — words that defer to a measure they never name, and feel definite precisely because the deferral is invisible. We reach for "random," or "uniform," or "typical," or "average," as though each pointed at one procedure the world would recognize, and each points at nothing until we quietly fix the thing to be spread evenly. The chord problem is a small machine for making that visible. It sets a word everyone trusts in front of a shape everyone can draw, and shows that the sentence held no answer of its own — that the answer was never waiting in the question, but in the choice we always make and seldom notice we are making: which of the countless things we might spread evenly, we do.