Here is a question: pick two points on a circle of radius \(r\) independently uniformly at random.

What is the average distance between these two points?

Analysis

Let's to analyze this question using probabilistic mathematical reasoning before doing simulations.

Sampling points on a circle

To sample a point \(P=(X …

Introduction

I am fascinated by the concept of Monte-Carlo simulation: you can, in principle, simulate random events and see how these simulations differ from reality (on a small scale, at least!)

Here I would like to show a couple of simple probability theory textbook problems and my attempt at simulating …