题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=4336
题意：买食品收集n个卡片，每个卡片的概率分别是pi，且Σp[i]<=1，求收集n个卡片需要买的食品数的期望。
压缩DP：把每个食品用二进制表示，0和1分别表示没有卡片和已经收集到此卡片的期望，则
f[s]=(1-Σp[i])*f[s]+Σp[j]*f[s]+Σp[k]*f[s|(1<<k)]
s表示状态，i表示所有卡片编号，j表示s状态中已经有的卡片编号，k表示s状态中没有的卡片编号
-> Σp[i]*f[s]=Σp[i]*f[s|(1<<i)]
或者容斥原理做：
压缩DP:
1 //STATUS:C++_AC_281MS_7128KB 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=(1<<20)+10; 37 const int INF=0x3f3f3f3f; 38 const int 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 p[23],f[N]; 59 int n; 60 61 int main(){ 62 // freopen("in.txt","r",stdin); 63 int i,j,up; 64 double s; 65 while(~scanf("%d",&n)) 66 { 67 for(i=0;i<n;i++){ 68 scanf("%lf",&p[i]); 69 } 70 up=(1<<n)-1; 71 f[up]=0; 72 for(i=up-1;i>=0;i--){ 73 f[i]=1;s=0; 74 for(j=0;j<n;j++){ 75 if(i&(1<<j))continue; 76 f[i]+=p[j]*f[i|(1<<j)]; 77 s+=p[j]; 78 } 79 f[i]/=s; 80 } 81 82 printf("%lf ",f[0]); 83 } 84 return 0; 85 }
容斥原理：
1 //STATUS:C++_AC_203MS_244KB 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=(1<<20)+10; 37 const int INF=0x3f3f3f3f; 38 const int 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 p[23]; 59 int n; 60 61 int main(){ 62 // freopen("in.txt","r",stdin); 63 int i,j,up,cnt; 64 double ans,s; 65 while(~scanf("%d",&n)) 66 { 67 for(i=0;i<n;i++){ 68 scanf("%lf",&p[i]); 69 } 70 up=(1<<n)-1;ans=0; 71 for(i=1;i<=up;i++){ 72 s=0; 73 for(j=cnt=0;j<n;j++){ 74 if(i&(1<<j)){ 75 cnt++; 76 s+=p[j]; 77 } 78 } 79 if(cnt&1)ans+=1/s; 80 else ans-=1/s; 81 } 82 83 printf("%lf ",ans); 84 } 85 return 0; 86 }