<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) {
    // When the game ends, the point is in [k..k - 1 maxPts].
    //   P = 1, if n >= k - 1 + maxPts
    //   P = 0, if n < k (note that the constraints already have k <= n)
    if (k == 0 || n >= k - 1 + maxPts)
      return 1.0;

    double ans = 0.0;
    vector<double> dp(n + 1);  // dp[i] := the probability to have i points
    dp[0] = 1.0;
    double windowSum = dp[0];  // P(i - 1) + P(i - 2) + ... + P(i - maxPts)

    for (int i = 1; i <= n; ++i) {
      // The probability to get i points is
      // P(i) = [P(i - 1) + P(i - 2) + ... + P(i - maxPts)] / maxPts
      dp[i] = windowSum / maxPts;
      if (i < k)
        windowSum += dp[i];
      else  // The game ends.
        ans += dp[i];
      if (i - maxPts >= 0)
        windowSum -= dp[i - maxPts];
    }

    return ans;
  }
};
11000