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):
1 //STATUS:C++_AC_15MS_320KB 2 #include <functional> 3 #include <algorithm> 4 #include <iostream> 5 //#include <ext/rope> 6 #include <fstream> 7 #include <sstream> 8 #include <iomanip> 9 #include <numeric> 10 #include <cstring> 11 #include <cassert> 12 #include <cstdio> 13 #include <string> 14 #include <vector> 15 #include <bitset> 16 #include <queue> 17 #include <stack> 18 #include <cmath> 19 #include <ctime> 20 #include <list> 21 #include <set> 22 #include <map> 23 using namespace std; 24 //#pragma comment(linker,"/STACK:102400000,102400000") 25 //using namespace __gnu_cxx; 26 //define 27 #define pii pair<int,int> 28 #define mem(a,b) memset(a,b,sizeof(a)) 29 #define lson l,mid,rt<<1 30 #define rson mid+1,r,rt<<1|1 31 #define PI acos(-1.0) 32 //typedef 33 typedef __int64 LL; 34 typedef unsigned __int64 ULL; 35 //const 36 const int N=130; 37 const int INF=0x3f3f3f3f; 38 const LL MOD=1000000007,STA=8000010; 39 const LL LNF=1LL<<55; 40 const double EPS=1e-9; 41 const double OO=1e30; 42 const int dx[4]={-1,0,1,0}; 43 const int dy[4]={0,1,0,-1}; 44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31}; 45 //Daily Use ... 46 inline int sign(double x){return (x>EPS)-(x<-EPS);} 47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;} 48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;} 49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;} 50 template<class T> inline T Min(T a,T b){return a<b?a:b;} 51 template<class T> inline T Max(T a,T b){return a>b?a:b;} 52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);} 53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);} 54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));} 55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));} 56 //End 57 58 double n,m; 59 60 int main(){ 61 // freopen("in.txt","r",stdin); 62 int i,j; 63 while(~scanf("%lf%lf",&n,&m)) 64 { 65 printf("%.10lf\n",n*(1-pow((n-1)/n,m))); 66 } 67 return 0; 68 }
O(m):
1 //STATUS:C++_AC_15MS_1675KB 2 #include <functional> 3 #include <algorithm> 4 #include <iostream> 5 //#include <ext/rope> 6 #include <fstream> 7 #include <sstream> 8 #include <iomanip> 9 #include <numeric> 10 #include <cstring> 11 #include <cassert> 12 #include <cstdio> 13 #include <string> 14 #include <vector> 15 #include <bitset> 16 #include <queue> 17 #include <stack> 18 #include <cmath> 19 #include <ctime> 20 #include <list> 21 #include <set> 22 #include <map> 23 using namespace std; 24 //#pragma comment(linker,"/STACK:102400000,102400000") 25 //using namespace __gnu_cxx; 26 //define 27 #define pii pair<int,int> 28 #define mem(a,b) memset(a,b,sizeof(a)) 29 #define lson l,mid,rt<<1 30 #define rson mid+1,r,rt<<1|1 31 #define PI acos(-1.0) 32 //typedef 33 typedef __int64 LL; 34 typedef unsigned __int64 ULL; 35 //const 36 const int N=100010; 37 const int INF=0x3f3f3f3f; 38 const LL MOD=1000000007,STA=8000010; 39 const LL LNF=1LL<<55; 40 const double EPS=1e-9; 41 const double OO=1e30; 42 const int dx[4]={-1,0,1,0}; 43 const int dy[4]={0,1,0,-1}; 44 const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31}; 45 //Daily Use ... 46 inline int sign(double x){return (x>EPS)-(x<-EPS);} 47 template<class T> T gcd(T a,T b){return b?gcd(b,a%b):a;} 48 template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;} 49 template<class T> inline T lcm(T a,T b,T d){return a/d*b;} 50 template<class T> inline T Min(T a,T b){return a<b?a:b;} 51 template<class T> inline T Max(T a,T b){return a>b?a:b;} 52 template<class T> inline T Min(T a,T b,T c){return min(min(a, b),c);} 53 template<class T> inline T Max(T a,T b,T c){return max(max(a, b),c);} 54 template<class T> inline T Min(T a,T b,T c,T d){return min(min(a, b),min(c,d));} 55 template<class T> inline T Max(T a,T b,T c,T d){return max(max(a, b),max(c,d));} 56 //End 57 58 double f[N]; 59 int n,m; 60 61 int main(){ 62 // freopen("in.txt","r",stdin); 63 int i,j; 64 double ans; 65 while(~scanf("%d%d",&n,&m)) 66 { 67 f[0]=1;ans=1; 68 for(i=1;i<m;i++){ 69 f[i]=(1-f[i-1])*f[i-1]+f[i-1]*(f[i-1]-1.0/n); 70 ans+=f[i]; 71 } 72 73 printf("%.10lf\n",ans); 74 } 75 return 0; 76 }