Bertrand’s Paradox Calculator — Three Answers
In 1889 Joseph Bertrand asked a question that sounds like it has one answer: draw a chord of a circle completely at random — what is the probability it is longer than the side of the inscribed equilateral triangle? It has three answers, and they are all correct. Choose how “random” will be done and this calculator runs the experiment twenty thousand times so you can watch each answer arrive at its own destination.
Three Procedures, Three Different Probabilities
The threshold first, before the procedures: a chord beats the triangle’s side when its midpoint lies within R/2 of the centre (the chord at distance d from the centre has length 2√(R² − d²), which exceeds √3 R exactly when d is less than R/2). Now the procedures diverge. Random endpoints: fix one endpoint, let the other fall anywhere on the circumference; the chord is long enough when the second point lands in the far third of the circle — probability 1/3. Random radius: choose a radius, then a uniformly random point along it as the midpoint; the winning midpoints fill the half of the radius nearer the centre — probability 1/2. Random midpoint: drop the midpoint uniformly over the disc; winners are those inside the concentric circle of radius R/2, whose area is (1/2)² = a quarter of the whole — probability 1/4. Three honest uniformities over three different spaces — arcs, lengths, areas — three answers. Bertrand’s lesson, which founded the modern insistence that a probability model names its randomisation, is that “at random” is not a specification; it is a family of them, and choosing the family is the modelling, done before any number appears.
Reading the Simulation’s Verdict
The widget’s draws are seeded (1889, the year of Bertrand’s Calcul des probabilités), so the experiment is repeatable to the chord — run it, note the percentage, switch methods, run again, and watch the estimate walk to a different neighbourhood with the same calm convergence. With 20,000 draws the sampling wobble is roughly a third of a percentage point, so the empirical figures shadow their theories closely: endpoints around 33–34, radius around fifty, midpoint in the mid-twenties. The simulation’s role is arbitration between intuitions, not between the answers: it cannot crown one method, because each method’s simulation is correct by construction, and a crown awarded by simulation to geometry would be exactly the category error Bertrand exposed. What it dissolves is the feeling that geometry owed us one answer — the circle never promised that; our language did.
Jaynes and the Principle of Indifference’s Repair
Twentieth-century opinion mostly followed E. T. Jaynes, who argued (1968) that a well-posed version of the problem should be indifferent to the circle’s position, size and orientation — throw straws of a given length at a circle drawn on the floor, from far enough away that no coordinate is privileged. Under those symmetries the random-radius answer, 1/2, is the unique survivor (his straw experiments, famously, agreed), and the other two procedures smuggle in coordinate privileges: uniform endpoints overweight long chords’ endpoints, uniform midpoints overweight the rim’s area. The calculator stays neutral — all three methods run, all three theories print — because the pedagogical gold is the dependence itself: Bertrand’s paradox is the cleanest exhibition ever staged of the fact that probability lives in the procedure, not in the object. Every applied statistician who has ever asked “random sample of what, drawn how?” is Bertrand’s heir.
The Buffon Cousin: Needles, Straws and Floors
Bertrand’s straws have an older, more famous relative: Buffon’s needle (1777), in which a needle dropped on floorboards crosses a crack with probability 2L ÷ πD — a formula so obligingly invertible that generations have estimated π by dropping needles and counting, the original Monte Carlo experiment. The family resemblance is the procedure-first moral: Buffon’s probability is well-defined because the dropping procedure is physically specified (uniform position, uniform angle — the symmetries Jaynes later demanded of Bertrand), and its π estimates inherit their honesty entirely from that specification. The simulation on this page is Buffon’s method in modern dress — a seeded, repeatable dropping of twenty thousand straws — and the two paradoxes together mark the twin births of geometric probability: one showing that randomness plus geometry computes constants of nature, the other showing that randomness unspecified computes nothing at all.
What a Chord Knows About Clinical Trials
The Bertrand discipline — name the mechanism before quoting the number — is the same one that rescues real studies from their favourite illusions. “Randomly selected patients” conceals a chord’s worth of ambiguity (the clinic’s walk-ins, the registry’s volunteers and the population’s true cross-section are three different uniforms over three different spaces), and treatment effects computed on the wrong one replicate Bertrand’s embarrassment at human scale. The paradox’s service to applied work is permanent vigilance disguised as a geometry game: every time a probability is quoted, someone should be able to answer “uniform over what, generated how?” as crisply as this page answers it for chords. When the answer is a shrug, the number that follows it is literature, not statistics — sometimes excellent literature, but not evidence.
Related Tools
More from the Statistics Calculators hub: its sibling Boy or Girl Paradox Calculator, where conditioning does the mischief, and the Two Envelopes Paradox Calculator — three paradoxes, three different joints in probability’s armour.
Frequently Asked Questions
So which answer is actually right, then?
Each — for its procedure — and the question as Bertrand posed it is under-specified, which is the discovery, not a dodge. If a physical process is supplied (straws thrown from afar: Jaynes’ 1/2; a spinner fixing two endpoints: 1/3; darts marking midpoints: 1/4), the answer is determined and the simulation reproduces it. Probability questions in the wild arrive with their procedures implied by context, and the paradox’s permanent service is to make the implication explicit — the same discipline as defining a sample space before computing anything, which is probability’s first commandment precisely because Bertrand showed what happens without it.
Why do the endpoints give 1/3, exactly?
Fix the first endpoint and draw the inscribed triangle on it: its other two vertices divide the circumference into three equal arcs. The chord from the fixed point beats the triangle’s side exactly when the second endpoint falls in the arc opposite — the far third — because only then does the chord span more than the triangle’s 120 degrees of arc. Uniform endpoints make each third equally likely by symmetry, so the answer is 1/3 with no integration at all. The elegance of this derivation is half the reason the endpoints method feels canonical to beginners — and Bertrand’s trap is that the other methods have derivations every bit as clean.
Why does the midpoint method give the smallest answer, then?
Area’s prejudice: midpoints uniformly scattered over the disc concentrate near the rim, because annuli grow with radius (the ring between 0.9R and R holds 19% of the disc’s area in its outer tenth). Winning midpoints must fall inside the small central circle of radius R/2 — a quarter of the area hosting all the victories — while the rim’s vast territory produces the short chords. Uniform-over-area is no more and no less natural than uniform-over-angle or uniform-over-radius; each uniformity is flat in its own coordinate and lumpy in the others, which is the deep reason no coordinate-free “random chord” exists.
Is Bertrand’s paradox related to Bertrand’s box paradox, then?
Only by author and by moral. The box paradox (three boxes, gold and silver coins) is a conditioning puzzle with a definite answer (2/3) that intuition botches — its sibling page on this hub computes it. The chord paradox is a specification puzzle whose answer genuinely depends on the randomisation. Bertrand built both in the same 1889 treatise to discipline a young probability theory that used “at random” and “equally likely” as if they were self-defining; the two puzzles attack the vagueness from opposite sides, which is why this hub keeps them as bookends.
Where does this matter outside geometry puzzles, then?
Everywhere a “random” sample’s frame is chosen lazily: survey sampling (random households versus random adults versus random phone numbers give three electorates), clinical trial allocation, Monte Carlo integration (the choice of sampling distribution is Bertrand’s choice in work clothes), and the stereology of materials and biology, where random sections through structures are literally Bertrand chords and the correction factors depend on the sectioning procedure. The paradox is a parlour game with an industrial pension — it collects royalties wherever uniformity is assumed instead of engineered.
How accurate is the simulation at 20,000 draws, then?
The standard error of a proportion near one-third at this size is about 0.33 percentage points, so repeated seeds agree to within roughly a point — tight enough to separate 25, 33 and 50 unambiguously, loose enough to remind the reader that simulation estimates rather than proves. The theorems (1/3, 1/2, 1/4) are proven in the sections above; the seeded run is their illustration, and its exact repeatability under seed 1889 is a deliberate nod to the paradox’s own demand: specify the procedure, and the numbers will behave.