Bertrand’s Chord Paradox, explained.
Bertrand’s chord paradox shows that different sampling measures can assign different probabilities to an informally described random chord.
Why it happens
Random endpoints, uniform radial midpoint distance and uniform midpoint area use different distance distributions. For a unit circle, chords exceed √3 when their midpoint distance is below one-half, giving probabilities one-third, one-half and one-quarter.
Bertrand’s chord paradox shows that different sampling measures can assign different probabilities to an informally described random chord.
Read the result
Length comparisons use the same unit circle and √3 threshold. Different midpoint-distance measures, rather than arithmetic errors, explain the protocol probabilities.
A worked example
Give random chord three precise meanings
Draw a batch under random endpoints and compare its long-chord fraction with one-third.
Switch to uniform radial midpoint distance and compare the new batch with one-half.
Switch to uniform midpoint area, where distance is square root of a uniform draw, and compare with one-quarter.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Random endpoints, uniform radial midpoint distance and uniform midpoint area use different distance distributions. For a unit circle, chords exceed √3 when their midpoint distance is below one-half, giving probabilities one-third, one-half and one-quarter.
Where this idea is useful
A practical use
Specify the data-generating protocol rather than assuming the word random uniquely defines a distribution.
A common misconception
“Calling a chord random already specifies its probability distribution.”
A randomisation instruction must define a sampling measure. Different physical or mathematical protocols can answer different precise questions.
What this explanation leaves out
- Each protocol is explicitly modelled. No method is presented as the uniquely correct interpretation of every physical chord-selection procedure.
Why take a square root for uniform midpoint area?
Disk area grows with squared radius, so uniform area corresponds to radius equal to the square root of a uniform unit draw.
Which sampling mechanism hides inside the word random in your question?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.