Thank you for visiting this site. This article covers “Bertrand’s Paradox.”
We use the phrase “choose at random” all the time, but it is not a single well-defined operation. With the same problem, simply changing the definition of “random” makes the answer 1/3, 1/2, or 1/4 — three different values.
The Problem
This problem was posed by French mathematician Joseph Bertrand in 1889.
A circle has an equilateral triangle inscribed in it. Draw a “random chord” of the circle. What is the probability that the chord is longer than a side of the inscribed equilateral triangle?
Intuitively, one answer seems to follow naturally. But considering three different methods of “drawing a random chord” yields three different answers.
Method 1: Random Endpoints (Answer: 1/3)
Fix one point on the circumference, then choose a second point on the circumference uniformly at random to form a chord.
Setting the fixed point at a vertex of the triangle, the chord is longer than a side of the triangle only when the other endpoint falls on the arc opposite the fixed vertex — which is 1/3 of the circumference.
Probability: 1/3.
Method 2: Random Midpoint on a Radius (Answer: 1/2)
Choose the midpoint of the chord uniformly at random along a radius of the circle.
A chord is longer than a side of the triangle when its midpoint falls within the inner half of the radius (closer to the centre). Choosing uniformly along the radius, the probability of satisfying this condition is 1/2.
Method 3: Random Midpoint in the Disk (Answer: 1/4)
Choose the midpoint of the chord uniformly at random from all points inside the circle.
A chord is longer than a side of the triangle when its midpoint falls inside a smaller circle centered at the centre of the original circle with radius equal to half the original. The area of the smaller circle is 1/4 of the larger, so the probability is 1/4.
The Same Problem, Different Answers
All three methods are solving the same problem — “draw a random chord in a circle.” Yet the answers are 1/3, 1/2, and 1/4 respectively.
Each calculation is internally correct. The issue is that the operation “draw a random chord” does not have a unique definition.
“Choose endpoints at random,” “choose a midpoint along a radius at random,” and “choose a midpoint in the disk at random” each define a different probability distribution, and they naturally yield different answers.
Is There a Correct Answer?
The standard view is that Bertrand’s Paradox has “no single correct answer,” though deeper arguments exist.
In 1973 mathematician E. T. Jaynes argued that physical symmetry requirements uniquely identify the “correct” probability distribution. His requirements are:
- Rotational invariance: The probability does not change when the circle is rotated.
- Scale invariance: The probability does not change when the circle is rescaled.
- Translational invariance: The probability does not change when the circle is moved.
The only chord-selection method satisfying all three is Method 3 (uniform midpoint in the disk), giving 1/4. Physical experiments dropping needles over a circle reportedly yield results close to this value.
However, this “solution” works by adding additional physical constraints to the problem, and in purely mathematical terms the problem remains indeterminate.
What This Paradox Teaches
Bertrand’s Paradox shows that in probability problems, failing to define “random” precisely makes the problem itself incomplete.
In everyday life, when we say “choose at random,” we implicitly assume some probability distribution. But that implicit assumption can vary from person to person, causing the same problem to yield different answers.
In modern statistics and data science this lesson is critical. When we say “sample users at random,” are we sampling from active users, all registered users, or with probability proportional to visit frequency? The results can differ dramatically. Bertrand’s Paradox showed 150 years ago that leaving “what exactly is random?” ambiguous leads to misinterpretation.
Summary
This article covered “Bertrand’s Paradox.”
“Random” is not an obvious, self-evident concept, as this paradox vividly demonstrates. Handling probability correctly requires defining assumptions precisely.
To return to the full list of paradoxes, follow the link below.
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