docs: public static double exactBinomial(int k, int n, double p) {
      double total = 0.0;
      for (int m = k; m <= n; m++) {
        double nChooseM = 1.0;
        for (int r = 1; r <= m; r++) {
          nChooseM *= (n - r) + 1;
          nChooseM /= r;
        }
        // System.out.println(n + " choose " + m + " is " + nChooseM);
        // System.out.println("prob contribution is " +
        //	       (nChooseM * Math.pow(p, m) * Math.pow(1.0-p, n - m)));
        total += nChooseM * Math.pow(p, m) * Math.pow(1.0 - p, n - m);
      }
      return total;
    }