← Back to Paradoxes

Bertrand's Paradox

Joseph Bertrand (1889) • The Ambiguity of "Random"

The Question

Draw a random chord on a circle.
What's the probability it's longer than the side of an inscribed equilateral triangle?

1/3 1/2 1/4

All three answers are correct! It depends on what "random" means.

Random Endpoints
Fix one endpoint on the circle. Choose the second endpoint uniformly at random on the circumference. The chord is long when the second point falls in a 120° arc.
Chords drawn 0
Long chords 0
Observed ratio 0%
Expected Probability
1/3 ≈ 33.3%
Random Radius Point
Choose a random point on a radius. Draw the chord perpendicular to the radius at that point. The chord is long when the point is in the inner half of the radius.
Chords drawn 0
Long chords 0
Observed ratio 0%
Expected Probability
1/2 = 50%
Random Midpoint
Choose a random point inside the circle. Draw the unique chord with this point as its midpoint. The chord is long when the point is inside an inner circle of radius r/2.
Chords drawn 0
Long chords 0
Observed ratio 0%
Expected Probability
1/4 = 25%

Side-by-Side Comparison

Method 1

Random point on circumference

1/3

Long arc is 120° of 360°

Method 2

Random point on radius

1/2

Inner half of radius works

Method 3

Random point in circle

1/4

Inner circle is 1/4 the area

Why Three Different Answers?

The phrase "random chord" is ambiguous! Each method defines a different probability distribution over the space of all possible chords.

In the 19th century, mathematicians assumed there was one "natural" answer. Bertrand showed this assumption was naive — without specifying how the randomness is generated, the question has no unique answer.

This paradox revolutionized probability theory by highlighting the need for precise problem specification. It's not enough to say "random" — you must say what is uniformly distributed.

Jaynes' Resolution (1973)

Physicist E.T. Jaynes argued that Method 2 is the "correct" answer if we add reasonable constraints:

  • Scale invariance: The answer shouldn't change if we resize the circle
  • Translation invariance: The answer shouldn't change if we move the circle
  • Rotation invariance: The answer shouldn't change if we rotate the circle

Only Method 2 satisfies all three! But without these constraints, all three methods remain valid.

Historical Note

French mathematician Joseph Bertrand presented this paradox in his 1889 book Calcul des probabilités. It caused a crisis in classical probability theory, which relied on the "principle of indifference" (if we have no reason to favor one outcome, all outcomes are equally likely). Bertrand showed that this principle is ambiguous when dealing with continuous distributions — there are multiple ways to be "equally likely," and they give different answers.