Bertrand's Paradox: Three Probabilities from One Question

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The radius of the circle. The probabilities are independent of the radius, but the chord and triangle lengths scale with it.
units
Choose which sampling procedure to focus on, or compare all three at once.
P - Method 1: Random endpointsThree valid answers co-exist: 1/3, 1/2, and 1/4
0.3333%

Two uniform random points on the circumference define the chord

Triangle side length17.3205units
P - Method 2: Random radial point0.5%
P - Method 3: Random midpoint0.25%
Critical distance (r/2)5units
Inner circle radius (Method 3)5units
Favorable arc (Method 1)120deg
Chord length at critical point17.3205units
Method 1: Endpoints (1/3)0.3333%
Method 2: Radial point (1/2)0.5%
Method 3: Midpoint (1/4)0.25%

Same question, three valid answers: 1/3, 1/2, and 1/4.

  • Method 1 (random endpoints, P = 1/3): Fix one endpoint; orient the triangle so a vertex coincides with it. The second endpoint produces a long chord only if it lands on the 120-degree arc opposite that vertex - exactly 1/3 of the full 360-degree circumference.
  • Method 2 (random radial point, P = 1/2): Pick a random point along a radius (uniform on [0, 10]). Draw the chord perpendicular at that point. The chord exceeds the side when the point is closer than r/2 = 5.0000 units to the center, which covers exactly half the radius length.
  • Method 3 (random midpoint, P = 1/4): Pick any point inside the disk uniformly. Use it as the chord midpoint. A long chord requires the midpoint inside the inner circle of radius 5.0000. That inner area is 78.54 sq units - exactly 1/4 of the disk area 314.16 sq units.
  • The triangle side is 17.3205 units for radius 10. The chord-length formula 2 * sqrt(r^2 - d^2) confirms that a chord is longer than this side if and only if its midpoint is within 5.0000 units of the center.

Next stepSelect a specific method from the dropdown to drill into that method's geometry and step-by-step derivation.

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