Bertrand's Paradox.

Same question. Three correct answers.
See randomness decide.

Pick a Method

Each method gives a different, mathematically correct probability.

Method 1

Long chord (> √3·r) Short chord
Trials
0
Favorable
0
Probability
0.000
Theoretical
0.333
Long chord iff both endpoints within 120° arc.

Compare all three

Why do three answers exist?

Bertrand (1889) asked: what is the probability that a random chord of a circle is longer than the side of the inscribed equilateral triangle?

Answer depends on what "random" means:

Method 1 — Random Endpoints. Pick two angles uniformly on the circle. Probability = 1/3.

Method 2 — Random Radius. Pick a radius, then a uniform point along it; chord is perpendicular. Probability = 1/2.

Method 3 — Random Midpoint. Midpoint uniform across the disk. Chord is long iff midpoint lies inside inner disk of radius 1/2. Area ratio = 1/4.

The "paradox" is that "uniform random chord" has no unique meaning without specifying the measure. Each method uses a different sample space.

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