2012 Asia Chengdu Regional Contest Problem B

Candy

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 1351    Accepted Submission(s): 586
Special Judge

Problem Description

LazyChild is a lazy child who likes candy very much. Despite being very young, he has two large candy boxes, each contains n candies initially. Everyday he chooses one box and open it. He chooses the first box with probability p and the second box with probability (1 - p). For the chosen box, if there are still candies in it, he eats one of them; otherwise, he will be sad and then open the other box.
He has been eating one candy a day for several days. But one day, when opening a box, he finds no candy left. Before opening the other box, he wants to know the expected number of candies left in the other box. Can you help him?

 

 

Input

There are several test cases.
For each test case, there is a single line containing an integer n (1 ≤ n ≤ 2 × 10⁵) and a real number p (0 ≤ p ≤ 1, with 6 digits after the decimal).
Input is terminated by EOF.

 

 

Output

For each test case, output one line “Case X: Y” where X is the test case number (starting from 1) and Y is a real number indicating the desired answer.
Any answer with an absolute error less than or equal to 10^-4 would be accepted.

 

 

Sample Input

10 0.400000 100 0.500000 124 0.432650 325 0.325100 532 0.487520 2276 0.720000

 

 

Sample Output

Case 1: 3.528175 Case 2: 10.326044 Case 3: 28.861945 Case 4: 167.965476 Case 5: 32.601816 Case 6: 1390.500000

 

 

Source

2012 Asia Chengdu Regional Contest

 

 

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#pragma comment(linker, "/STACK:1024000000,1024000000")
#include <map>
#include <queue>
#include <vector>
#include <string>
#include <cmath>
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;
#define maxn 10005
#define mod 1000000007
#define ll long long
#define INF 0x7fffffff
int n, m;
int main(){
    int cas = 1;
    int t;
    /*scanf("%d", &t);
    while (t--){
        
    }*/
    double p;
    while (~scanf("%d%lf", &n,&p)){
        double p1 = log(p);
        double p2 = log(1 - p);
        double s1 = (n + 1)*p1;
        double s2 = (n + 1)*p2;
        double ans = 0,c=0;
        for (int i = 0; i < n; i++){
            if(c+s1>-30||c+s2>-30)ans += (exp(c+s1) + exp(c+s2))*(n - i);
            c += log(n + i + 1) - log(i + 1);
            s1 = s1 + p2;
            s2 = s2 + p1;
        }
        printf("Case %d: ", cas++);
        printf("%.6lf\n", ans);
    }
    return 0;
}