<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:

    int maxPts, diff, N;
    vector<double> dp;
    vector<double> dp2;

    double getSuffix(int k) {
        if(k < 0 && abs(k) > N) return 0;
        if(dp2[k + N] != -1.0) return dp2[k + N];
        return dp2[k + N] = helper(k) + getSuffix(k - 1);
    }

    double helper(int k){ 
        if(k <= 0) return (double)(abs(k) <= diff);
        double ans = 0;
        ans = getSuffix(k - 1) - getSuffix((k - maxPts) - 1);
        ans = (ans/(double)maxPts);
        return ans;
    }

    double new21Game(int n, int k, int mpts) {
       maxPts = mpts;
       diff = n - k;
       N = n;
       dp2 = vector<double>(n + k + 1, -1.0);
       return helper(k);
    }
};
14800