Codeforces Round #293 (Div. 2) D. Ilya and Escalator

D. Ilya and Escalator

time limit per test
2 seconds

memory limit per test
256 megabytes

input
standard input

output
standard output

Ilya got tired of sports programming, left university and got a job in the subway. He was given the task to determine the escalator load factor.

Let's assume that n people stand in the queue for the escalator. At each second one of the two following possibilities takes place: either
the first person in the queue enters the escalator with probability p, or the first person in the queue doesn't move with probability (1 - p),
paralyzed by his fear of escalators and making the whole queue wait behind him.

Formally speaking, the i-th person in the queue cannot enter the escalator until people with indices from 1 to i - 1 inclusive
enter it. In one second only one person can enter the escalator. The escalator is infinite, so if a person enters it, he never leaves it, that is he will be standing on the escalator at any following second. Ilya needs to count the expected value of the number
of people standing on the escalator after t seconds.

Your task is to help him solve this complicated task.

Input

The first line of the input contains three numbers n, p, t (1 ≤ n, t ≤ 2000, 0 ≤ p ≤ 1).
Numbers n and t are
integers, number p is real, given with exactly two digits after the decimal point.

Output

Print a single real number — the expected number of people who will be standing on the escalator after t seconds. The absolute or relative
error mustn't exceed 10 - 6.

Sample test(s)

input

1 0.50 1

output

0.5

input

1 0.50 4

output

0.9375

input

4 0.20 2

output

0.4

概率dp

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <set>
#include <queue>

using namespace std;
#define maxn 2005
#define inf 0x7ffffff
double dp[maxn][maxn]; //dp[i][j] 表示第i秒电梯上有j个人的概率
int n,t;
double p;
int main()
{
while(scanf("%d%lf%d",&n,&p,&t) != EOF){
memset(dp,0,sizeof(dp));
dp[0][0] = 1;
for(int i = 1; i <= t; i++){
for(int j = n; j >= 0;j--){
if(j == n){
dp[i][j] = dp[i-1][j-1] * p + dp[i-1][j];
}else if(j != 0){
dp[i][j] = dp[i-1][j-1] * p + dp[i-1][j] *(1-p);
}else{
dp[i][j] = dp[i-1][j] *(1 - p);
}
}
}
double ans = 0.0;
for(int i = 0;i <= t;i++){
ans += dp[t][i] *i;
}
printf("%.7lf\n",ans);
}
return 0;
}