100 random chords, how many intersections?

2026-06-16 · Source: 3Blue1Brown · Field: Science & Research — Mathematics & Computational Sciences · Depth: Intermediate, quick

Summary

A mathematical puzzle challenges readers to determine the expected number of intersection points formed by N random chords drawn within a circle, specifically asking for the cases of 10 and 100 such chords. The problem emphasizes the importance of precisely defining a "random chord," referencing Bertrand's Paradox. For this puzzle, a random chord is defined by selecting two distinct points uniformly on the circle, where the probability of a point landing in a given arc is proportional to that arc's length, and then connecting these two chosen points. This clarification is essential for accurately approaching the calculation of expected intersections.

Key takeaway

For mathematicians or problem-solvers tackling geometric probability, understanding the precise definition of "random" is paramount. If you are approaching problems involving "random chords" or similar geometric constructions, ensure you clarify the underlying probability distribution for point selection, as different interpretations can lead to vastly different results, as exemplified by Bertrand's Paradox. This foundational clarity prevents misinterpretations and ensures accurate calculations.

Key insights

Precisely defining "random" is critical for geometric probability problems, such as calculating expected intersections of chords on a circle.

Principles

• "Random" definitions are crucial in geometric probability.
• Point selection probability is proportional to arc length.

Method

A random chord is formed by choosing a first point uniformly on a circle, then a second point uniformly on the circle, and connecting them.

Topics

• Geometric Probability
• Random Chords
• Bertrand's Paradox
• Expected Value
• Circle Geometry
• Mathematical Puzzles

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Editorial summary, takeaway, and curation by AIssential. Original article published by 3Blue1Brown.