HDU 5810 Balls and Boxes

Balls and Boxes

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 291 Accepted Submission(s): 213


[align=left]Problem Description[/align]
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=∑mi=1(Xi−X¯)2m

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.

[align=left]Input[/align]
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.

[align=left]Output[/align]
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.

[align=left]Sample Input[/align]

2
1
2
2
0
0

[align=left]Sample Output[/align]

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.

球落入每个盒子的概率是1/m

概率不变，n个球 可以当作n次实验 可用二项分布做

方差 D(x)=n*p*q

p=1/m q=1-1/m

带入可得 D(x)=n*(m-1)/(m*m)

方差的期望就是 方差的均值 而每个盒子的方差是相等的，所以期望的值就是方差的值了

/* ***********************************************
Author        :guanjun
Created Time  :2016/8/10 12:46:03
File Name     :hdu5810.cpp
************************************************ */
#include <iostream>
#include <cstring>
#include <cstdlib>
#include <stdio.h>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <string>
#include <math.h>
#include <stdlib.h>
#include <iomanip>
#include <list>
#include <deque>
#include <stack>
#define ull unsigned long long
#define ll long long
#define mod 90001
#define INF 0x3f3f3f3f
#define maxn 10010
#define cle(a) memset(a,0,sizeof(a))
const ull inf = 1LL << 61;
const double eps=1e-5;
using namespace std;
priority_queue<int,vector<int>,greater<int> >pq;
struct Node{
int x,y;
};
struct cmp{
bool operator()(Node a,Node b){
if(a.x==b.x) return a.y> b.y;
return a.x>b.x;
}
};

bool cmp(int a,int b){
return a>b;
}
int main()
{
#ifndef ONLINE_JUDGE
//freopen("in.txt","r",stdin);
#endif
//freopen("out.txt","w",stdout);
ll n,m;
while(scanf("%lld %lld",&n,&m)!=EOF){
if(n==0&&m==0)break;
ll a=n*(m-1);
ll b=m*m;
ll k=__gcd(a,b);
printf("%lld/%lld\n",a/k,b/k);
}
return 0;
}