ZOJ 3329 One Person Game(概率DP)
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题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3329
题意:有三个骰子,分别有k1,k2,k3个面,每次掷骰子,初始分数为0,如果三个面分别为a,b,c则分数置0,否则加上三个骰子的分数之和,当分数大于n时结束。求游戏的期望步数
思路:期望一般逆推,概率一般正推,思路出自这里的题解,写得很好:http://blog.csdn.net/morgan_xww/article/details/6775853
设 E[i]表示现在分数为i,到结束游戏所要掷骰子的次数的期望值。
显然 E[>n] = 0; E[0]即为所求答案;
E[i] = ∑Pk*E[i+k] + P0*E[0] + 1; (Pk表示点数和为k的概率,P0表示分数清零的概率)
由上式发现每个 E[i]都包含 E[0],而 E[0]又是我们要求的,是个定值。
设 E[i] = a[i]*E[0] + b[i];
将其带入上面的式子:
E[i] = ( ∑Pk*a[i+k] + P0 )*E[0] + ∑Pk*b[i+k] + 1;
显然,
a[i] = ∑Pk*a[i+k] + P0;
b[i] = ∑Pk*b[i+k] + 1;
当 i > n 时:
E[i] = a[i]*E[0] + b[i] = 0;
所以 a[i>n] = b[i>n] = 0;
可依次算出 a[n],b[n]; a[n-1],b[n-1] ... a[0],b[0];
则 E[0] = b[0]/(1 - a[0]);
#include <cstdlib>#include <cctype>#include <cstring>#include <cstdio>#include <cmath>#include <algorithm>#include <vector>#include <string>#include <iostream>#include <map>#include <set>#include <queue>#include <stack>#include <bitset>#include <functional>#include <utility>using namespace std;const int maxn = 100010;const int maxm = 200020;const int inf = 0x3f3f3f3f;double x[600], y[600];double p[100];int main(){ int t; scanf("%d", &t); while(t--) { int n, k1, k2, k3, a, b, c; scanf("%d%d%d%d%d%d%d", &n, &k1, &k2, &k3, &a, &b, &c); double p0 = 1.0 / k1 / k2 / k3; memset(p, 0, sizeof(p)); for(int i = 1; i <= k1; i++) for(int j = 1; j <= k2; j++) for(int k = 1; k <= k3; k++) if(i != a || j != b || k != c) p[i + j + k] += p0; memset(x, 0, sizeof(x)); memset(y, 0, sizeof(y)); for(int i = n; i >= 0; i--) { for(int j = 1; j <= k1 + k2 + k3; j++) { x[i] += x[i + j] * p[j]; y[i] += y[i + j] * p[j]; } x[i] += p0; y[i] += 1; } printf("%.16lf\n", y[0] / (1 - x[0])); } return 0;}
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