<p>Alice plays the following game, loosely based on the card game <strong>&quot;21&quot;</strong>.</p>

<p>Alice starts with <code>0</code> points and draws numbers while she has less than <code>k</code> points. During each draw, she gains an integer number of points randomly from the range <code>[1, maxPts]</code>, where <code>maxPts</code> is an integer. Each draw is independent and the outcomes have equal probabilities.</p>

<p>Alice stops drawing numbers when she gets <code>k</code> <strong>or more points</strong>.</p>

<p>Return the probability that Alice has <code>n</code> or fewer points.</p>

<p>Answers within <code>10<sup>-5</sup></code> of the actual answer are considered accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 10, k = 1, maxPts = 10
<strong>Output:</strong> 1.00000
<strong>Explanation:</strong> Alice gets a single card, then stops.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 6, k = 1, maxPts = 10
<strong>Output:</strong> 0.60000
<strong>Explanation:</strong> Alice gets a single card, then stops.
In 6 out of 10 possibilities, she is at or below 6 points.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 21, k = 17, maxPts = 10
<strong>Output:</strong> 0.73278
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>0 &lt;= k &lt;= n &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= maxPts &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution {
 public:
  double new21Game(int n, int k, int maxPts) {
    if (k == 0)
      return 1.0;

    vector<double> dp(k);
    dp[0] = 1.0;
    double baseProbability{1.0 / maxPts};
    double windowSum{1.0};

    for (int i = 1; i < k; i++) {
      dp[i] = windowSum * baseProbability;
      windowSum += dp[i];

      if (i >= maxPts)
        windowSum -= dp[i - maxPts];
    }

    double no{0.0};
    double de{0.0};

    for (int i = 1; k - i >= 0 && i <= maxPts; i++) {
      int endCards{maxPts - i + 1};
      int validCards{min(maxPts, n - k + i) - i + 1};
      no += dp[k - i] * validCards;
      de += dp[k - i] * endCards;
    }

    return no / de;
  }

 private:
  int H(int n, int k) {
    int i{n + k - 1};
    int j{min(n - 1, k)};

    int output{1};
    for (int m = 1; m <= j; m++) {
      output *= i - j + m;
      output /= m;
    }

    return output;
  }
};
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