Calculate probabilities for the number of bits to set in a uniformly random bit string to simulate a higher density of bits; see https://math.stackexchange.com/questions/2830624

Question2830624.java

import java.math.BigInteger;

public class Question2830624 {
	final static double p = 0.5;
	final static double q = 0.6;
	final static int n = 20;

	public static void main (String [] args) {
		double [] a = new double [n + 1];
		double [] b = new double [n + 1];		
		double [] f = new double [n + 1];

		for (int k = 0;k <= n;k++) {
			a [k] = binomialCoefficient (n,k) * Math.pow (p,k) * Math.pow (1 - p,n - k);
			b [k] = binomialCoefficient (n,k) * Math.pow (q,k) * Math.pow (1 - q,n - k);
		}

		for (int k = 0;k <= n;k++) {
			double [] c = new double [k + 1];
			for (int i = 0;i <= k;i++)
				for (int j = 0;j <= i;j++)
					c [i] += a [k - j] * binomialCoefficient (n - k + j,j) * binomialCoefficient (k - j,i - j) / binomialCoefficient (n,i);
			f [k] = b [k];
			for (int i = 0;i < k;i++)
				f [k] -= c [i] * f [i];
			f [k] /= c [k];
		}

		double sum = 0;
		
		for (int k = 0;k <= n;k++) {
			System.out.println (k + " " + f [k]);
			sum += f [k];
		}
		
		System.out.println ();
		System.out.println ("sum: " + sum);
	}
	
	static BigInteger factorial (long n) {
		return n == 0 ? BigInteger.ONE : factorial (n - 1).multiply (BigInteger.valueOf (n));
	}

	static double binomialCoefficient (long n,long k) {
		return factorial (n).divide (factorial (k)).divide (factorial (n - k)).doubleValue ();
	}
}