Bertrand’s paradox illustrates that the probability of a random chord being longer than the side of an inscribed equilateral triangle depends on how “randomness” is defined. The classic problem asks: given a circle of radius R, what is the chance that a chord drawn at random exceeds the length sqrt{3},R?
Three common constructions lead to three distinct probabilities. The first selects two random points on the circumference and draws the chord; the second chooses a random radius and a point uniformly along it; the third picks a random point inside the circle as the chord’s midpoint. Each method yields a different distribution of chord lengths.
Using the chord‑length formula, the condition L > sqrt{3},R translates to a threshold on theta. Depending on the random‑generation method, the probability that theta exceeds this threshold is 1/3, 1/2, or 1/4, which is the essence of Bertrand’s paradox.
What is Bertrand's paradox?
How many common constructions are there for this problem?
What is the length of the side of an inscribed equilateral triangle in terms of the circle's radius R?
How does this calculator help solve Bertrand's paradox?
What are the three distinct probabilities mentioned in the paradox?
Can this calculator be used for any other geometric probability problems?
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