<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:
    def new21Game(self, n: int, k: int, maxPts: int) -> float:
        # the maximum pts at k - 1 position can achieve 
        # k - 1 + maxPts <= n then probability is one
        # if n >= k -1 + maxPts:
        #     return 1
        dp = [0] * (n + 1)
        dp[0] = 1
        run = 1 # prefix sum 
        for i in range(1, n+1):
            dp[i] = run/maxPts
            if i < k:
                run += dp[i]
            if i >= maxPts and i - maxPts < k:
                run -= dp[i-maxPts]
            # sump = 0
            # for j in range(1, maxPts + 1):
            #     if i - j < 0:
            #         break
            #     if i - j < k:
            #         sump += dp[i - j] * (1/maxPts)
            # dp[i] = sump
        
        # print(dp)
        res = 0
        for i in range(k, n+1):
            res += dp[i]
        # print('res', res)
        return min(res, 1)
16700