Probably one of the least efficient way to approximate e.

MonteCarloEuler.java

import java.util.Random;

public class MonteCarloEuler
{
    private static final Random random = new Random();

    public static void main(String[] args)
    {
        final int total = 100000;
        final int n = 10000;
        int successes = 0;

        for(int i = 0; i < total; i++)
            if(!binomialTrial(n))
                successes++;

        System.out.println((double) total / successes);
    }

    private static boolean binomialTrial(int n)
    {
        for(int i = 0; i < n; i++)
            if(random.nextInt(n) == 0)
                return true;
        return false;
    }
}