Approximating Euler’s Number Using a Random Number Generator

I stumbled upon a mathematical curiosity on Twitter:

Pick random numbers between 0 and 1, until the sum is greater than 1. Repeat forever. The average number of numbers picked will be Euler’s number (e=2.718281828…)

Here is the proof: twitter.com/fermatslibrary/status/961238636418293763

The kids wanted to try, so we coded it in C#:

	using System;
	namespace EulerNumberGenerator
	{
	class Program
	{
	static Random rand = new Random();
	static void Main(string[] args)
	{
	double sum = 0;
	int amount = int.MaxValue;
	for (int i = 0; i < amount; i++)
	{
	int picks = DoRandomPicks();
	sum = sum + picks;
	double avg = sum / (i + 1);
	Console.WriteLine("Avg:" + avg + " Error: " + (avg – Math.E));
	}
	}
	static int DoRandomPicks()
	{
	int pickCounter = 0;
	double sum = 0;
	while (sum < 1)
	{
	double r = rand.NextDouble();
	sum = sum + r;
	pickCounter++;
	}
	return pickCounter;
	}
	}
	}

view raw

EulerNumberGenerator.cs

hosted with ❤ by GitHub

 

Here is a snippet of output:

Avg:2.70660032651792 Error:-0.0116815019411254
Avg:2.70662313432836 Error:-0.0116586941306869
Avg:2.70664593859308 Error:-0.0116358898659632
Avg:2.70666873931292 Error:-0.0116130891461275
Avg:2.70669153648869 Error:-0.0115902919703532

After a few thousand iterations the error went down to around 0.0001

It then occurred to me that this function might be an easy way to test the randomness of a Random Number Generator.

What do you think?