Compare long-chord probabilities under three different randomization procedures.
Mathematical Curiosities & Special Topics
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1EnterProvide the known values
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Use the labelled probability protocol, game rule, finite bound, or numerical model. Simulations are seeded and reproducible; theoretical results remain separate. Inputs stay on this device.
Long-chord probability
Calculated resultCalculated from the values currently entered
Understand and verify
Probability chord exceeds inscribed-triangle side:0.333333333333
Randomization method:Choose two independent uniform endpoints on the circumference
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A mathematical curiosity makes a surprising pattern, process, game, approximation, or paradox explicit enough to explore step by step.
Use it to explore the named mathematical model with visible assumptions rather than treating a surprising output as a universal claim.
The relationship
Write the rule before calculating
Two independent uniform circumference endpoints give 1/3; uniform orientation plus uniform perpendicular distance from 0 to R gives 1/2; a midpoint uniform by disk area gives 1/4.
See the structure
What the calculation is doing
startapply rulenext staterepeatobserve patternTwo independent uniform circumference endpoints give 1/3; uniform orientation plus uniform perpendicular distance from 0 to R gives 1/2; a midpoint uniform by disk area gives 1/4.
Worked interpretation
Read the result in context
The answer changes because 'random chord' is not one probability model.
Interpret with care
Important boundary
Explorers use the stated finite inputs, simulation limits, and conventions; a result should not be generalized beyond that model.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Select the stated protocol, strategy, approximation, or conversion mode when one is available.
Enter the labelled finite bound, probabilities, payoffs, sequence state, or special-function argument.
Read the theoretical or deterministic result separately from any seeded empirical result.
Calculation method
Apply the stated mathematical model
Two independent uniform circumference endpoints give 1/3; uniform orientation plus uniform perpendicular distance from 0 to R gives 1/2; a midpoint uniform by disk area gives 1/4.
The calculator validates model hypotheses, finite work limits, probability domains, singular cases, and numerical approximation ranges before returning a result.
Worked example
Worked example
The answer changes because 'random chord' is not one probability model.
Two independent uniform circumference endpoints give 1/3; uniform orientation plus uniform perpendicular distance from 0 to R gives 1/2; a midpoint uniform by disk area gives 1/4.
Supported inputs
Precision and limits
Model scope
Paradoxes depend on their information or randomization protocol. Queueing, games, fractals, and cellular automata use exactly the visible assumptions; changing those assumptions changes the answer.
Deterministic versus simulated
Exact combinatorial and theoretical probabilities are labelled separately from seeded pseudorandom simulations. Reusing a seed reproduces the same finite experiment.
Numerical methods
Finite double-precision arithmetic is used for probability models and special functions. Gamma uses a bounded Lanczos approximation, Bessel functions use bounded convergent series and order limits, and erf/erfc disclose their approximation error.
Limits
Iteration, grid, candidate, trial, and series caps are visible in field labels or calculator-specific notes. A bounded trace never claims to prove an unresolved conjecture.
Calculator-specific rule
Each mode displays its complete probability measure; choosing only a radius orientation does not define a chord.
