3Blue1Brown
June 16, 2026
Expected Intersections of Random Chords on a Circle
TL;DR
A mathematical puzzle exploring how many intersection points are expected when 10 or 100 random chords are drawn on a circle, building on Bertrand's paradox.
Key Takeaways
- 1Bertrand's paradox highlights the ambiguity in defining 'random chords' on a circle
- 2A random chord is precisely defined by choosing two uniformly distributed points on the circle's circumference
- 3The expected number of intersection points can be calculated using probabilistic methods for different numbers of chords
- 4The puzzle connects to a series of related mathematical problems across consecutive months
Notable Quotes
“I hear some of you say, I remember something fishy about choosing random chords on a circle. That is right. There's a famous paradox here called Bertrand's paradox.”
— Presenter
“When I say choose a random chord, I mean start by choosing one point uniformly on the circle.”
— Presenter
Chapters
1. Introduction to the Puzzle
The presenter introduces the main puzzle: calculating the expected number of intersection points for 10 and 100 random chords on a circle.
2. Bertrand's Paradox and Precision
Discussion of Bertrand's paradox and the importance of precisely defining what 'random chord' means to avoid ambiguity.
3. Defining Random Chord Selection
Clear mathematical definition: choose two points uniformly at random on the circle's circumference and connect them.
4. The Challenge Ahead
The presenter poses the computational challenge and hints at connections to previous puzzles in the series.
Key People & Entities
- Numberphile
- YouTube channel that previously covered Bertrand's paradox in detail
Glossary
- Bertrand's Paradox
- A famous paradox in probability theory demonstrating that the answer to 'what is the probability a random chord is longer than the side of an inscribed equilateral triangle' depends on how 'random chord' is defined
- Uniform Distribution
- A probability distribution where all outcomes within a given range are equally likely; here, every point on the circle has equal probability of being chosen
- Expected Value
- The average outcome of a random process; in this case, the predicted number of intersection points on average