<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 find(int p , int n , int k , int maxi , vector <double> &dp , vector <double> &pref)
    {
        if(p>=k)
        {
            if(p<=n) return 1.0 ;
            else return 0.0 ;
        }
        if(dp[p] != -1.0) return dp[p] ;
        double ans = (1.0/maxi)*(pref[n]-pref[p]) ;
        for(int i = 1 ; i<=maxi ; i++)
        {
            ans += (1.0/maxi)*find(p+i , n , k , maxi,dp,pref) ;
        }
        return ans ;
    }
    double new21Game(int n, int k, int maxPts) 
    {
        vector <double> pref(n+1,0.0) ;
        for(int i = n ; i>=k ; i--)
        {
            pref[i] = 1.0 ;
            if(i<n) pref[i] += pref[i+1] ;
        }
        vector <double> dp(n+1,0.0) ;
        for(int i = k ; i<=n ; i++) dp[i] = 1.0 ;
        for(int i = k-1 ; i>=0 ; i--)
        {
            //double ans = 0.0 ;
            int endindex = min(i+maxPts,n) ;
            dp[i] = pref[i+1] ;
            if(endindex<n) dp[i] -= pref[endindex+1] ;
            dp[i] *= (1.0/maxPts) ;
            pref[i] = dp[i] + pref[i+1] ;
            // for(int j = 1 ; j<=maxPts ; j++)
            // {
            //     if(i+j>n) break ;
            //     ans += (dp[i+j]) ;
            // }
            //dp[i] = (1.0/maxPts)*(ans) ;
        }
        return dp[0] ;
    }
};
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