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📐 Bertrand's Paradox

One Question, Three Valid Answers

The Question (Bertrand, 1888)

A chord is drawn at random in a circle with an inscribed equilateral triangle.
What is the probability that the chord is longer than a side of the triangle?

Longer than side
Shorter than side
Triangle side
100ms

📊 Statistics

Chords longer 0
Chords shorter 0
Total chords 0
Observed Probability
-
Expected: 1/3 ≈ 33.3%

Method 1: Random Endpoints

Pick two random points on the circle's circumference and draw a chord between them. The chord is longer than the triangle side only if the second point falls in the far 120° arc (1/3 of the circle).

🤯 Three Methods, Three Answers!

Bertrand showed that this simple geometric question has three equally valid answers depending on how you interpret "random chord":

Method 1: Random Endpoints

1/3

Pick two random points on the circle. Only 1/3 of the arc produces long chords.

Method 2: Random Radius

1/2

Pick a random point on a radius as the chord's midpoint. Half are in the inner region.

Method 3: Random Midpoint

1/4

Pick a random point inside the circle. Only 1/4 fall in the inner circle (radius R/2).

💡 The Profound Lesson

"Random" is meaningless without specifying HOW you randomize!

All three methods are mathematically correct. The paradox reveals that probability depends on our model of randomness, not just the geometric setup.

In 1973, physicist Edwin Jaynes proposed a resolution: if you physically throw straws at a circle from far away, the "random radius" method (P = 1/2) emerges naturally. But this just picks one physical interpretation — the mathematical ambiguity remains.

The deeper message: When someone says "pick a random X," always ask: "Random according to what distribution?"