Alice plays the following game, loosely based on the card game "21".
Alice starts with 0 points, and draws numbers while she has less than K points.  During each draw, she gains an integer number of points randomly from the range [1, W], where W is an integer.  Each draw is independent and the outcomes have equal probabilities.
Alice stops drawing numbers when she gets K or more points.  What is the probability that she has N or less points?
Example 1:
Input: N = 10, K = 1, W = 10
Output: 1.00000
Explanation:  Alice gets a single card, then stops.

Example 2:
Input: N = 6, K = 1, W = 10
Output: 0.60000
Explanation:  Alice gets a single card, then stops.
In 6 out of W = 10 possibilities, she is at or below N = 6 points.

Example 3:
Input: N = 21, K = 17, W = 10
Output: 0.73278
Note:

0 <= K <= N <= 10000
1 <= W <= 10000
Answers will be accepted as correct if they are within 10^-5 of the correct answer.
The judging time limit has been reduced for this question.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n    def new21Game(self, N: int, K: int, W: int) -> float:\n        \n        # solution 1: Time and Space both O(N+W)\n        # # pseudocode: let dp[x] be the answer when\n        # #             Alice already has x points\n        # dp[k] = 1.0 when K <= k <= N, else 0.0\n        # for k from K-1 to 0:\n        #     dp[k] = (dp[k+1] + ... + dp[k+W]) / W\n        # return dp[0]\n        \n        # dp[x] = the answer when Alice has x points\n        dp = [0.0] * (N + W + 1)\n        for k in range(K, N + 1):\n            # Alice wins if has between K and N points\n            dp[k] = 1.0\n\n        S = sum(dp[i] for i in range(K, K+W+1))#min(N - K + 1, W) # = dp[K+1] + ... + dp[K+W]\n        # S = dp[k+1] + dp[k+2] + ... + dp[k+W]\n        for k in reversed(list(range(K))):\n            dp[k] = S / float(W)\n            S += dp[k] - dp[k + W]\n\n        return dp[0]\n        \n#         # solution 2: time and space limit exceeded\n#         # dp(n, k) = 1 / W * ( sum pt in range(1, W+1) over dp(n-pt, k-pt) )\n        \n#         memo = {}\n#         def dp(n, k):\n#             if (n, k) not in memo:\n#                 if k <= 0:\n#                     ans = int(n >= 0)\n#                 else:\n#                     ans = 0\n#                     for pt in range(1, W+1):\n#                         ans += dp(n-pt, k-pt)\n#                     ans *= 1. / W\n#                 memo[n, k] = ans\n#             return memo[n, k]\n#         return dp(N, K)\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'new21Game'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6100