HDU5810-Balls and Boxes

最新推荐文章于 2019-08-21 10:53:13 发布

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最新推荐文章于 2019-08-21 10:53:13 发布

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探讨n个球随机放入m个盒子后的球数方差期望值，通过数学推导得出简洁公式，并附带高效算法实现。

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Balls and Boxes

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 868 Accepted Submission(s): 556

Problem Description

Mr. Chopsticks is interested in random phenomena, and he conducts an experiment to study randomness. In the experiment, he throws n balls into m boxes in such a manner that each ball has equal probability of going to each boxes. After the experiment, he calculated the statistical variance V as

V = \sum m i = 1 ( X i - X ¯ ) 2 m

where

Xi is the number of balls in the ith box, and

X¯ is the average number of balls in a box.
Your task is to find out the expected value of V.

Input

The input contains multiple test cases. Each case contains two integers n and m (1 <= n, m <= 1000 000 000) in a line.
The input is terminated by n = m = 0.

Output

For each case, output the result as A/B in a line, where A/B should be an irreducible fraction. Let B=1 if the result is an integer.

Sample Input

Sample Output

  
   0/1
1/2


   
    
     Hint
    
In the second sample, there are four possible outcomes, two outcomes with V = 0 and two outcomes with V = 1.

Author

SYSU

Source

2016 Multi-University Training Contest 7

题意：n个球，放到m个盒子里，问每个盒子球个数的方差的均值

解题思路：打表找规律后发现ans = n*(m-1) / m^2

#include <iostream>
#include <cstdio>
#include <cstring>
#include <string>
#include <algorithm>
#include <cmath>
#include <map>
#include <cmath>
#include <set>
#include <stack>
#include <queue>
#include <vector>
#include <bitset>
#include <functional>

using namespace std;

#define LL long long
const int INF = 0x3f3f3f3f;

LL n,m;

LL gcd(LL x,LL y)
{
    return x?gcd(y%x,x):y;
}

int main()
{
    while(~scanf("%lld%lld",&n,&m)&&(n+m))
    {
        if(m==1) {printf("0/1\n");continue;}
        n=(m-1)*n;
        m=m*m;
        printf("%lld/%lld\n",n/gcd(n,m),m/gcd(n,m));
    }
    return 0;
}