BZOJ 4008 亚瑟王(概率DP 奥妙重重)

题意

中文题面，就不解释了

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分析

显然这道题直接求期望太麻烦，想想转化问题（这转化太神了）。
定义$f(i,j)$表示第$i$张卡总共被经过$j$次的概率，有转移方程式
$\large f(i,j)=f(i-1,j)*(1-p_{i-1})^j+f(i-1,j+1)*(1-(1-p_{i-1})^{j+1})$最终答案就是
$\large \sum_{i=1}^n\sum_{j=1}^rf(i,j)*(1-(1-p[i])^j)*d_i$

$-.-$

AC CODE

#include <bits/stdc++.h>
using namespace std;
const int MAXN = 250;
const int MAXM = 150; 
double p[MAXN], d[MAXN], f[MAXN][MAXM], mul[MAXN][MAXM];
int main () {
  int T, n, m;
  scanf("%d", &T);
  while(T--) {
    scanf("%d%d", &n, &m);
    for(int i = 1; i <= n; ++i) {
      scanf("%lf%lf", &p[i], &d[i]);
      mul[i][0] = 1;
      for(int j = 1; j <= m; ++j)
        mul[i][j] = mul[i][j-1] * (1-p[i]);
    }
    memset(f, 0, sizeof f);
    f[1][m] = 1; double ans = 0;
    for(int i = 1; i <= n; ++i)
      for(int j = m; j >= 1; --j) {
        if(!(i == 1 && j == m))
          f[i][j] = f[i-1][j] * mul[i-1][j] + f[i-1][j+1] * (1-mul[i-1][j+1]);
        ans += f[i][j] * (1-mul[i][j]) * d[i];
      }
    printf("%.10lf\n", ans);
  }
}

TIP:预处理幂快的多