期望 ——kuangbin A - Collecting Bugs POJ - 2096

返回二级目录——kaugnbin概率dp习题集

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原题

原题传送门（私有）

Ivan is fond of collecting. Unlike other people who collect post stamps, coins or other material stuff, he collects software bugs. When Ivan gets a new program, he classifies all possible bugs into n categories. Each day he discovers exactly one bug in the program and adds information about it and its category into a spreadsheet. When he finds bugs in all bug categories, he calls the program disgusting, publishes this spreadsheet on his home page, and forgets completely about the program. 
Two companies, Macrosoft and Microhard are in tight competition. Microhard wants to decrease sales of one Macrosoft program. They hire Ivan to prove that the program in question is disgusting. However, Ivan has a complicated problem. This new program has s subcomponents, and finding bugs of all types in each subcomponent would take too long before the target could be reached. So Ivan and Microhard agreed to use a simpler criteria --- Ivan should find at least one bug in each subsystem and at least one bug of each category. 
Macrosoft knows about these plans and it wants to estimate the time that is required for Ivan to call its program disgusting. It's important because the company releases a new version soon, so it can correct its plans and release it quicker. Nobody would be interested in Ivan's opinion about the reliability of the obsolete version. 
A bug found in the program can be of any category with equal probability. Similarly, the bug can be found in any given subsystem with equal probability. Any particular bug cannot belong to two different categories or happen simultaneously in two different subsystems. The number of bugs in the program is almost infinite, so the probability of finding a new bug of some category in some subsystem does not reduce after finding any number of bugs of that category in that subsystem. 
Find an average time (in days of Ivan's work) required to name the program disgusting.

Input

Input file contains two integer numbers, n and s (0 < n, s <= 1 000).

Output

Output the expectation of the Ivan's working days needed to call the program disgusting, accurate to 4 digits after the decimal point.

Sample Input

1 2

Sample Output

3.0000

题意

题目大意：s个系统，n种bug，每种bug出现概率相同都为1/n,每种bug等概率出现在每个系统中都为1/s，每个系统有无限个bug，每天找到一个bug，在保证每个系统都至少有一个bug的前提下，求找全n种bug的期望天数。

思路

期望倒着推，因为期望已知条件为e(end)=0

dp[i][j]表示当前已经有i个系统有bug，找到了j类bug，dp值为从当前状态转移到dp[s][n]的期望，则dp[s]{n]=0;

有 dp[i][j] =dp[i][j+1]*i*(n-j)/(s*n) + dp[i+1][j]*(s-i)*j/(s*n)+dp[i+1][j+1]*(s-i)*(n-j)/(s*n)+dp[i][j]*(i*j)/(s*n) + 1;

将dp[i][j]移到一边解方程得最终公式：

dp[i][j]=dp[i][j+1]*i*(n-j)/(s*n-i*j) + dp[i+1][j]*(s-i)*j/(s*n-i*j) + dp[i+1][j+1]*(s-i)*(n-j)/(s*n-i*j)+ 1.0*s*n/(s*n-i*j)

AC代码

#include
using namespace std;
double dp[1002][1002];
int main(){
    int n,s;
    cin>>n>>s;
    //dp[i][j]=dp[i][j+1]*i*(n-j)/(s*n-i*j) + dp[i+1][j]*(s-i)*j/(s*n-i*j) + dp[i+1][j+1]*(s-i)*(n-j)/(s*n-i*j)+ 1.0*s*n/(s*n-i*j)
    //dp[s]{n]=0;
    for(int i=s;i>=0;i--){
        for(int j=n;j>=0;j--)
            if(i!=s || j!=n)
                dp[i][j]=dp[i][j+1]*i*(n-j)/(s*n-i*j) + 
                    dp[i+1][j]*(s-i)*j/(s*n-i*j)+
                    dp[i+1][j+1]*(s-i)*(n-j)/(s*n-i*j)+
                    1.0*s*n/(s*n-i*j);
    }
    printf("%.6f\n",dp[0][0]);
}