docs: public static double approximateLowerBoundOnP(final long n, final long k, final double numStdDevs) {
      checkInputs(n, k);
      if (n == 0) { return 0.0; } // the coin was never flipped, so we know nothing
      else if (k == 0) { return 0.0; }
      else if (k == 1) { return (exactLowerBoundOnPForKequalsOne(n, deltaOfNumStdevs(numStdDevs))); }
      else if (k == n) { return (exactLowerBoundOnPForKequalsN(n, deltaOfNumStdevs(numStdDevs))); }
      else {
        final double x = abramowitzStegunFormula26p5p22((n - k) + 1, k, (-1.0 * numStdDevs));
        return (1.0 - x); // which is p
      }
    }