CODEWITHSUNDEEP
javamathrandomprobability
sup4eli
sup4eli

Reputation: 324

Comparing two methods of choosing random numbers in [-1, +1]

I was taking an online java course and given the following challenge:

Compute an estimate of π by simulating throwing darts onto the unit square and determining the fraction of them that lie in the unit circle.

My idea for how to generate the random x coordinate was to pick two uniformly-random numbers in the range [0, 1], then subtract them. The resulting values then range from -1 to +1.

The instructor answer was to multiply the generated double by 2 and subtract 1. Here's their code:

public class MonteCarlo
{
public static void main(String[] args)
{
    System.out.println("Number of tries");
    Random generator = new Random(42);
    Scanner in = new Scanner(System.in);
    int tries = in.nextInt();

    int hits = 0;
    for (int i = 1; i <= tries; i++)
    {
        // Generate two random numbers between -1 and 1

        double x = generator.nextDouble() * 2 -1 ;

        double y = generator.nextDouble() * 2 -1 ;

        // Check whether the point lies in the unit circle

        if (Math.sqrt((x - 0) * (x - 0) + (y - 0) * (y - 0)) <= 1)
        {
            hits++;
        }
    }

    /*
       The ratio hits / tries is approximately the same as the ratio
       circle area / square area = pi / 4
    */

    double piEstimate = 4.0 * hits / tries;
    System.out.println("Estimate for pi: " + piEstimate);
}
}

My question is, how do my instructor's approach and my approach differ? Does my idea work?

Upvotes: 1

Views: 280

Answers (1)

templatetypedef
templatetypedef

Reputation: 373032

This is actually more of a math question than a programming question. Your goal is to sample a uniformly-random real number in the range [-1, 1], and you have two ways of doing this:

  1. Sample a real number in the range [0, 1], multiply it by two, and subtract one.

  2. Sample two real numbers in the range [0, 1] uniformly, then subtract them.

Method (1) works correctly. Method (2) does not. Rather than explaining it for the range [0, 1], let's imagine instead you want to pick a random number between -5 and +5 by rolling two fair dice and subtracting the first die from the second. Even though every die roll is fair and uniformly picks something out of {1, 2, 3, 4, 5, 6}, the resulting distribution isn't uniform. For example:

  • The odds of getting -5 are one in thirty-six, because there's only one possible way for it to happen: you have to roll a one on the first die and a six on the second die.
  • The odds of getting 0 are one in six, because there are six different rolls that produce it (1/1, 2/2, 3/3, 4/4, 5/5, and 6/6).

The same mathematical sort of argument works out to explain why subtracting two uniformly-random numbers won't give you a uniformly-random distribution over the range you want. Instead, you'll get a triangularly-shaped distribution that's peaked at 0 and drops off as you go towards +1 and -1.

So in other words, you do get a random number - it just isn't equivalent to throwing a dart at the unit square with uniform probability.

Upvotes: 3

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