UVA11427 Expect the Expected
题目大意:每晚打游戏。每晚中,赢一局概率p,最多玩n局,一直玩到胜率严格大于p时才能停止游戏并高高兴兴回家,否则只能打满n局垂头丧气回家。问打游戏的天数的期望。
考虑“垂头丧气回家”的概率
设dp[i][j]表示i天,赢了j天,且胜率小于等于p的概率
dp[i][j] = dp[i-1][j] * (1 - p) + dp[i - 1][j - 1] * p,j/i <= p
初始状态dp[0][0] = 1,dp[0][i] = 0,i > 0
Q = dp[n][0] + dp[n][1] + .... + dp[n][n]
局数期望EX = Q + 2Q(1-Q) + 3Q(1-Q)^2 + ... + iQ(1-Q)^(i-1) + ... 这是一个无穷级数
运用数列中错位相减求和的思想,有
EX = Q + 2Q(1-Q) + 3Q(1-Q)^2 + ... + iQ(1-Q)^(i-1) + ...
(1-Q)EX = Q(1-Q) + 2Q(1-Q)^2 + ... + (i-1)Q(1-Q)^(i-1) + ......
两式相减,有EX = 1 + (1-Q) + (1-Q)^2 + (1-Q)^3 + .... + (1-Q0^i +........ = lim(i->∞) (1 + (1 - Q)^i)/(1 - (1- Q)) = 1/Q
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 #include <cstdlib> 5 #include <algorithm> 6 #include <queue> 7 #include <vector> 8 #include <cmath> 9 #define min(a, b) ((a) < (b) ? (a) : (b)) 10 #define max(a, b) ((a) > (b) ? (a) : (b)) 11 #define abs(a) ((a) < 0 ? (-1 * (a)) : (a)) 12 inline void swap(int &a, int &b) 13 { 14 long long tmp = a;a = b;b = tmp; 15 } 16 inline void read(int &x) 17 { 18 x = 0;char ch = getchar(), c = ch; 19 while(ch < '0' || ch > '9') c = ch, ch = getchar(); 20 while(ch <= '9' && ch >= '0') x = x * 10 + ch - '0', ch = getchar(); 21 if(c == '-') x = -x; 22 } 23 24 const int INF = 0x3f3f3f3f; 25 const int MAXN = 100 + 10; 26 27 int t,px,py,n,ca; 28 double dp[MAXN][MAXN]; 29 30 int main() 31 { 32 read(t); 33 for(ca = 1;t;--t,++ca) 34 { 35 read(px), read(py), read(n); 36 memset(dp, 0, sizeof(dp)); 37 dp[0][0] = 1; 38 for(register int i = 1;i <= n;++ i) 39 for(register int j = 0;j <= n;++ j) 40 { 41 if(py * j <= px * i) 42 { 43 if(j - 1 >= 0) dp[i][j] = dp[i - 1][j] * (1 - (double)px / py) + dp[i - 1][j - 1] * ((double)px / py); 44 else dp[i][j] = dp[i - 1][j] * (1 - (double)px / py); 45 } 46 else 47 { 48 dp[i][j] = 0; 49 continue; 50 } 51 } 52 double sum = 0; 53 for(register int i = 0;i <= n;++ i) sum += dp[n][i]; 54 int ans = 1 / sum; 55 printf("Case #%d: %d\n", ca, ans); 56 } 57 return 0; 58 }