Question

837. New 21 Game

中文文档

Description

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, maxPts], where maxPts is an integer. Each draw is independent and the outcomes have equal probabilities.

Alice stops drawing numbers when she gets k or more points.

Return the probability that Alice has n or fewer points.

Answers within 10-5 of the actual answer are considered accepted.

 

Example 1:

Input: n = 10, k = 1, maxPts = 10
Output: 1.00000
Explanation: Alice gets a single card, then stops.

Example 2:

Input: n = 6, k = 1, maxPts = 10
Output: 0.60000
Explanation: Alice gets a single card, then stops.
In 6 out of 10 possibilities, she is at or below 6 points.

Example 3:

Input: n = 21, k = 17, maxPts = 10
Output: 0.73278

 

Constraints:

• 0 <= k <= n <= 104
• 1 <= maxPts <= 104

Solutions

Solution 1

class Solution:
  def new21Game(self, n: int, k: int, maxPts: int) -> float:
    @cache
    def dfs(i: int) -> float:
      if i >= k:
        return int(i <= n)
      if i == k - 1:
        return min(n - k + 1, maxPts) / maxPts
      return dfs(i + 1) + (dfs(i + 1) - dfs(i + maxPts + 1)) / maxPts

    return dfs(0)

class Solution {
  private double[] f;
  private int n, k, maxPts;

  public double new21Game(int n, int k, int maxPts) {
    f = new double[k];
    this.n = n;
    this.k = k;
    this.maxPts = maxPts;
    return dfs(0);
  }

  private double dfs(int i) {
    if (i >= k) {
      return i <= n ? 1 : 0;
    }
    if (i == k - 1) {
      return Math.min(n - k + 1, maxPts) * 1.0 / maxPts;
    }
    if (f[i] != 0) {
      return f[i];
    }
    return f[i] = dfs(i + 1) + (dfs(i + 1) - dfs(i + maxPts + 1)) / maxPts;
  }
}

class Solution {
public:
  double new21Game(int n, int k, int maxPts) {
    vector<double> f(k);
    function<double(int)> dfs = [&](int i) -> double {
      if (i >= k) {
        return i <= n ? 1 : 0;
      }
      if (i == k - 1) {
        return min(n - k + 1, maxPts) * 1.0 / maxPts;
      }
      if (f[i]) {
        return f[i];
      }
      return f[i] = dfs(i + 1) + (dfs(i + 1) - dfs(i + maxPts + 1)) / maxPts;
    };
    return dfs(0);
  }
};

func new21Game(n int, k int, maxPts int) float64 {
  f := make([]float64, k)
  var dfs func(int) float64
  dfs = func(i int) float64 {
    if i >= k {
      if i <= n {
        return 1
      }
      return 0
    }
    if i == k-1 {
      return float64(min(n-k+1, maxPts)) / float64(maxPts)
    }
    if f[i] > 0 {
      return f[i]
    }
    f[i] = dfs(i+1) + (dfs(i+1)-dfs(i+maxPts+1))/float64(maxPts)
    return f[i]
  }
  return dfs(0)
}

function new21Game(n: number, k: number, maxPts: number): number {
  const f = new Array(k).fill(0);
  const dfs = (i: number): number => {
    if (i >= k) {
      return i <= n ? 1 : 0;
    }
    if (i === k - 1) {
      return Math.min(n - k + 1, maxPts) / maxPts;
    }
    if (f[i] !== 0) {
      return f[i];
    }
    return (f[i] = dfs(i + 1) + (dfs(i + 1) - dfs(i + maxPts + 1)) / maxPts);
  };
  return dfs(0);
}

Solution 2

class Solution:
  def new21Game(self, n: int, k: int, maxPts: int) -> float:
    f = [0] * (k + maxPts)
    for i in range(k, min(n + 1, k + maxPts)):
      f[i] = 1
    f[k - 1] = min(n - k + 1, maxPts) / maxPts
    for i in range(k - 2, -1, -1):
      f[i] = f[i + 1] + (f[i + 1] - f[i + maxPts + 1]) / maxPts
    return f[0]

class Solution {
  public double new21Game(int n, int k, int maxPts) {
    if (k == 0) {
      return 1.0;
    }
    double[] f = new double[k + maxPts];
    for (int i = k; i < Math.min(n + 1, k + maxPts); ++i) {
      f[i] = 1;
    }
    f[k - 1] = Math.min(n - k + 1, maxPts) * 1.0 / maxPts;
    for (int i = k - 2; i >= 0; --i) {
      f[i] = f[i + 1] + (f[i + 1] - f[i + maxPts + 1]) / maxPts;
    }
    return f[0];
  }
}

class Solution {
public:
  double new21Game(int n, int k, int maxPts) {
    if (k == 0) {
      return 1.0;
    }
    double f[k + maxPts];
    memset(f, 0, sizeof(f));
    for (int i = k; i < min(n + 1, k + maxPts); ++i) {
      f[i] = 1;
    }
    f[k - 1] = min(n - k + 1, maxPts) * 1.0 / maxPts;
    for (int i = k - 2; i >= 0; --i) {
      f[i] = f[i + 1] + (f[i + 1] - f[i + maxPts + 1]) / maxPts;
    }
    return f[0];
  }
};

func new21Game(n int, k int, maxPts int) float64 {
  if k == 0 {
    return 1
  }
  f := make([]float64, k+maxPts)
  for i := k; i < min(n+1, k+maxPts); i++ {
    f[i] = 1
  }
  f[k-1] = float64(min(n-k+1, maxPts)) / float64(maxPts)
  for i := k - 2; i >= 0; i-- {
    f[i] = f[i+1] + (f[i+1]-f[i+maxPts+1])/float64(maxPts)
  }
  return f[0]
}

function new21Game(n: number, k: number, maxPts: number): number {
  if (k === 0) {
    return 1;
  }
  const f = new Array(k + maxPts).fill(0);
  for (let i = k; i < Math.min(n + 1, k + maxPts); ++i) {
    f[i] = 1;
  }
  f[k - 1] = Math.min(n - k + 1, maxPts) / maxPts;
  for (let i = k - 2; i >= 0; --i) {
    f[i] = f[i + 1] + (f[i + 1] - f[i + maxPts + 1]) / maxPts;
  }
  return f[0];
}