SGU-495 Kids and Prizes 概率DP

　　题目链接：http://acm.sgu.ru/problem.php?contest=0&problem=495

　　题意：有n个盒子，每个盒子里面放了一个奖品，m个人轮流去选择盒子，如果盒子里面有奖品，就把奖品拿走，盒子留下，否则直接走人。求最后被抽走的奖品数期望。。。

　　如果直接考虑用期望来建立DP，复杂度都很高，而且题目精度过高。换一个角度思考，考虑每个奖品不被拿走的概率(1/n)^m，那么不被拿走的期望就是n*(1/n)^m，则拿走的期望n-n*(1/n)^m，复杂度O(1)。还可以考虑f[i]表示第 i 个人拿到奖品的概率，那么1-f[i]就是没有拿到奖品的概率，是对立事件，则f[i]=(1-f[i-1])*f[i-1] + f[i-1]*(f[i-1]-1/n)，(1-f[i-1])*f[i-1]表示第 j-1 个人没有拿到奖品，f[i-1]*(f[i-1]-1/n)表示第 j-1 个人拿到奖品了，那么接下来拿到奖品的概率就会减小1/n。。

　　O(1):

 //STATUS:C++_AC_15MS_320KB
 #include <functional>
 #include <algorithm>
 #include <iostream>
 //#include <ext/rope>
 #include <fstream>
 #include <sstream>
 #include <iomanip>
 #include <numeric>
 #include <cstring>
 #include <cassert>
 #include <cstdio>
 #include <string>
 #include <vector>
 #include <bitset>
 #include <queue>
 #include <stack>
 #include <cmath>
 #include <ctime>
 #include <list>
 #include <set>
 #include <map>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,102400000")
 //using namespace __gnu_cxx;
 //define
 #define pii pair<int,int>
 #define mem(a,b) memset(a,b,sizeof(a))
 #define lson l,mid,rt<<1
 #define rson mid+1,r,rt<<1|1
 #define PI acos(-1.0)
 //typedef
 typedef __int64 LL;
 typedef unsigned __int64 ULL;
 //const
 const int N=;
 const int INF=0x3f3f3f3f;
 const LL MOD=,STA=;
 const LL LNF=1LL<<;
 const double EPS=1e-;
 const double OO=1e30;
 const int dx[]={-,,,};
 const int dy[]={,,,-};
 const int day[]={,,,,,,,,,,,,};
 //Daily Use ...
 inline int sign(double x){return (x>EPS)-(x<-EPS);}
 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}
 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}
 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}
 template<class T> inline T Min(T a,T b){return a<b?a:b;}
 template<class T> inline T Max(T a,T b){return a>b?a:b;}
 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}
 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}
 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}
 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}
 //End

 double n,m;

 int main(){
   //  freopen("in.txt","r",stdin);
     int i,j;
     while(~scanf("%lf%lf",&n,&m))
     {
         printf("%.10lf\n",n*(-pow((n-)/n,m)));
     }
     return ;
 }

　　O(m):

 //STATUS:C++_AC_15MS_1675KB
 #include <functional>
 #include <algorithm>
 #include <iostream>
 //#include <ext/rope>
 #include <fstream>
 #include <sstream>
 #include <iomanip>
 #include <numeric>
 #include <cstring>
 #include <cassert>
 #include <cstdio>
 #include <string>
 #include <vector>
 #include <bitset>
 #include <queue>
 #include <stack>
 #include <cmath>
 #include <ctime>
 #include <list>
 #include <set>
 #include <map>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,102400000")
 //using namespace __gnu_cxx;
 //define
 #define pii pair<int,int>
 #define mem(a,b) memset(a,b,sizeof(a))
 #define lson l,mid,rt<<1
 #define rson mid+1,r,rt<<1|1
 #define PI acos(-1.0)
 //typedef
 typedef __int64 LL;
 typedef unsigned __int64 ULL;
 //const
 const int N=;
 const int INF=0x3f3f3f3f;
 const LL MOD=,STA=;
 const LL LNF=1LL<<;
 const double EPS=1e-;
 const double OO=1e30;
 const int dx[]={-,,,};
 const int dy[]={,,,-};
 const int day[]={,,,,,,,,,,,,};
 //Daily Use ...
 inline int sign(double x){return (x>EPS)-(x<-EPS);}
 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;}
 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}
 template<class T> inline T lcm(T a,T b,T d){return a/d*b;}
 template<class T> inline T Min(T a,T b){return a<b?a:b;}
 template<class T> inline T Max(T a,T b){return a>b?a:b;}
 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);}
 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);}
 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));}
 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));}
 //End

 double f[N];
 int n,m;

 int main(){
  //   freopen("in.txt","r",stdin);
     int i,j;
     double ans;
     while(~scanf("%d%d",&n,&m))
     {
         f[]=;ans=;
         for(i=;i<m;i++){
             f[i]=(-f[i-])*f[i-]+f[i-]*(f[i-]-1.0/n);
             ans+=f[i];
         }

         printf("%.10lf\n",ans);
     }
     return ;
 }