---
title: '100 Random Chords: How Many Intersections?'
source: 'https://youtube.com/watch?v=wGffBCfrAsE'
video_id: 'wGffBCfrAsE'
date: 2026-08-10
duration_sec: 70
---

# 100 Random Chords: How Many Intersections?

> Source: [100 Random Chords: How Many Intersections?](https://youtube.com/watch?v=wGffBCfrAsE)

## Summary

This video explores a probability puzzle: given 10 or 100 randomly chosen chords on a circle, what is the expected number of intersection points inside the circle? The presenter clarifies the definition of 'random chord' by specifying that each endpoint is chosen uniformly on the circle, and references Bertrand's paradox to highlight the importance of this definition.

### Key Points

- **The Puzzle** [00:01] — The video poses the question: if you choose 10 random chords on a circle, what is the expected number of intersection points inside the circle? It then extends the question to 100 chords.
- **Bertrand's Paradox Reference** [00:15] — The presenter notes that choosing random chords on a circle is famously associated with Bertrand's paradox, which shows that different definitions of 'random' can lead to different answers.
- **Definition of Random Chord** [00:29] — To avoid ambiguity, the presenter defines a random chord as one where both endpoints are chosen uniformly on the circle. This means the probability of a point landing in a given arc is proportional to that arc's length.
- **Connection to Previous Puzzles** [00:58] — The presenter mentions that this puzzle is part of a series of puzzles from the last two months, suggesting a thematic connection.

### Conclusion

The video sets up a clear probability puzzle about chord intersections, emphasizing the importance of precise definitions in probability problems. The answer to the expected number of intersections is not provided in the transcript, leaving the viewer to solve it or await a follow-up.

## Transcript

Imagine you choose 10 random chords on a circle. The puzzle is, what's the expected number of intersection points inside that circle? And what about if instead it was 100 [music] randomly chosen chords?
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. of Numberphile videos. So, let me be a little bit more precise here. When I say
choose a random chord, I mean start by choosing one point uniformly on the circle. Loosely speaking, that means every point is equally likely. If you're precise, it really means the probability that a point lands in a given arc is
proportional to that arc's length. Then choose a second point the same way and mean when I say choose a random chord. So again, imagining you choose 10 such number of intersection points inside that circle?
that circle? And what if it was 100 such chords? is a reason that this puzzle and the ones from the last 2 months were put ones from the last 2 months were put together.