SRM 504.5(2-1000pt)

DIV2 1000pt

题意：一群人排队，每次操作由要骰子决定，只要没有人中奖，游戏就不结束。若摇骰子摇出4，则队列第一个人中奖；否则，若摇的是奇数，则第一个人排队到队伍末尾去；否则，第一个人出局。若游戏途中，队列只剩一个人，则直接中奖。若摇了k次骰子仍然没人中奖，则此时队列第一个人自动获奖，游戏结束。给出k，问队列初始时为n个人，此时排在第m个的人中奖的概率有多大。n, m, k <= 10。

解法：普通的概率dp。将题中的k记为num。

　　　法一：设d[i][j][k]表示当前状态为摇了i次骰子，队列共有j人，排在第k个的中奖的概率。具体状态转移方程见代码，很简单。

　　　法二：设d[i][j][k]表示由初始状态变成摇了i次骰子后，队列有j个人，初始时排在第m个的人现在排在第k个的概率。状态转移方程同见方程。

tag:概率dp

法一：

 // BEGIN CUT HERE
 /*
  * Author:  plum rain
  * score :
  */
 /*

  */
 // END CUT HERE
 #line 11 "TheTicketsDivTwo.cpp"
 #include <sstream>
 #include <stdexcept>
 #include <functional>
 #include <iomanip>
 #include <numeric>
 #include <fstream>
 #include <cctype>
 #include <iostream>
 #include <cstdio>
 #include <vector>
 #include <cstring>
 #include <cmath>
 #include <algorithm>
 #include <cstdlib>
 #include <set>
 #include <queue>
 #include <bitset>
 #include <list>
 #include <string>
 #include <utility>
 #include <map>
 #include <ctime>
 #include <stack>

 using namespace std;

 #define CLR(x) memset(x, 0, sizeof(x))
 #define CLR1(x) memset(x, -1, sizeof(x))
 #define PB push_back
 #define SZ(v) ((int)(v).size())
 #define zero(x) (((x)>0?(x):-(x))<eps)
 #define out(x) cout<<#x<<":"<<(x)<<endl
 #define tst(a) cout<<#a<<endl
 #define CINBEQUICKER std::ios::sync_with_stdio(false)

 typedef vector<int> VI;
 typedef vector<string> VS;
 typedef vector<double> VD;
 typedef long long int64;

 const double eps = 1e-;
 const double PI = atan(1.0)*;
 const int maxint = ;

 double d[][][];

 class TheTicketsDivTwo
 {
     public:
         double find(int n, int m, int num){
             double t1 = (double)  / (double), t2 = (double) / (double);
             CLR (d);
             for (int i = ; i <= n; ++ i)
                 d[num][i][] = ;

             for (int i = num-; i >= ; -- i)
                 for (int j = ; j <= n; ++ j)
                     for (int k = ; k <= j; ++ k){
                         if (j == ) d[i][j][k] = ;
                         else if (k == ) d[i][j][k] = d[i+][j][j]*0.5 + t2;
                         else d[i][j][k] = d[i+][j][k-]*0.5 + d[i+][j-][k-]*t1;
                     }
             return d[][n][m];
         }

 // BEGIN CUT HERE
     public:
     void run_test(int Case) { if ((Case == -) || (Case == )) test_case_0(); if ((Case == -) || (Case == )) test_case_1(); if ((Case == -) || (Case == )) test_case_2(); if ((Case == -) || (Case == )) test_case_3(); }
     private:
     template <typename T> string print_array(const vector<T> &V) { ostringstream os; os << "{ "; for (typename vector<T>::const_iterator iter = V.begin(); iter != V.end(); ++iter) os << '\"' << *iter << "\","; os << " }"; return os.str(); }
     void verify_case(int Case, const double &Expected, const double &Received) { cerr << "Test Case #" << Case << "..."; if (Expected == Received) cerr << "PASSED" << endl; else { cerr << "FAILED" << endl; cerr << "\tExpected: \"" << Expected << '\"' << endl; cerr << "\tReceived: \"" << Received << '\"' << endl; } }
     void test_case_0() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.16666666666666666; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_1() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.5833333333333334; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_2() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.0; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_3() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.25264033564814814; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }

 // END CUT HERE

 };

 // BEGIN CUT HERE
 int main()
 {
 //    freopen( "a.out" , "w" , stdout );
     TheTicketsDivTwo ___test;
     ___test.run_test(-);
        return ;
 }
 // END CUT HERE

法二：

 // BEGIN CUT HERE
 /*
  * Author:  plum rain
  * score :
  */
 /*

  */
 // END CUT HERE
 #line 11 "TheTicketsDivTwo.cpp"
 #include <sstream>
 #include <stdexcept>
 #include <functional>
 #include <iomanip>
 #include <numeric>
 #include <fstream>
 #include <cctype>
 #include <iostream>
 #include <cstdio>
 #include <vector>
 #include <cstring>
 #include <cmath>
 #include <algorithm>
 #include <cstdlib>
 #include <set>
 #include <queue>
 #include <bitset>
 #include <list>
 #include <string>
 #include <utility>
 #include <map>
 #include <ctime>
 #include <stack>

 using namespace std;

 #define CLR(x) memset(x, 0, sizeof(x))
 #define CLR1(x) memset(x, -1, sizeof(x))
 #define PB push_back
 #define SZ(v) ((int)(v).size())
 #define zero(x) (((x)>0?(x):-(x))<eps)
 #define out(x) cout<<#x<<":"<<(x)<<endl
 #define tst(a) cout<<#a<<endl
 #define CINBEQUICKER std::ios::sync_with_stdio(false)

 typedef vector<int> VI;
 typedef vector<string> VS;
 typedef vector<double> VD;
 typedef long long int64;

 const double eps = 1e-;
 const double PI = atan(1.0)*;
 const int maxint = ;

 double d[][][];

 class TheTicketsDivTwo
 {
     public:
         double find(int n, int m, int num){
             double t1 = (double)  / (double), t2 = (double) / (double);
             CLR (d);
             d[][n][m] = 1.0;
             for (int i = ; i <= num; ++ i)
                 for (int j = ; j <= n; ++ j)
                     for (int k = ; k <=j; ++ k){
                         if (j == k){
                             if (j == ) d[i][][] = d[i-][j+][j+] * t1;
                             else d[i][j][k] = d[i-][j][] * 0.5 + d[i-][j+][j+] * t1;
                         }
                         else{
                             d[i][j][k] = d[i-][j][k+] * 0.5;
                             if (j < n) d[i][j][k] += d[i-][j+][k+] * t1;
                         }
                     }

             double ans = ;
             for (int i = ; i < num; ++ i)
                 for (int j = ; j <= n; ++ j)
                     ans += d[i][j][] * (j ==  ?  : t2);
             for (int i = ; i <= n; ++ i)
                 ans += d[num][i][];
             return ans;
         }

 // BEGIN CUT HERE
     public:
     void run_test(int Case) { if ((Case == -) || (Case == )) test_case_0(); if ((Case == -) || (Case == )) test_case_1(); if ((Case == -) || (Case == )) test_case_2(); if ((Case == -) || (Case == )) test_case_3(); }
     private:
     template <typename T> string print_array(const vector<T> &V) { ostringstream os; os << "{ "; for (typename vector<T>::const_iterator iter = V.begin(); iter != V.end(); ++iter) os << '\"' << *iter << "\","; os << " }"; return os.str(); }
     void verify_case(int Case, const double &Expected, const double &Received) { cerr << "Test Case #" << Case << "..."; if (Expected == Received) cerr << "PASSED" << endl; else { cerr << "FAILED" << endl; cerr << "\tExpected: \"" << Expected << '\"' << endl; cerr << "\tReceived: \"" << Received << '\"' << endl; } }
     void test_case_0() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.16666666666666666; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_1() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.5833333333333334; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_2() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.0; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }
     void test_case_3() { int Arg0 = ; int Arg1 = ; int Arg2 = ; double Arg3 = 0.25264033564814814; verify_case(, Arg3, find(Arg0, Arg1, Arg2)); }

 // END CUT HERE

 };

 // BEGIN CUT HERE
 int main()
 {
 //    freopen( "a.out" , "w" , stdout );
     TheTicketsDivTwo ___test;
     ___test.run_test(-);
        return ;
 }
 // END CUT HERE