SGU495Kids and Prizes（数学期望||概率DP||公式）

 

495. Kids and Prizes

Time limit per test: 0.25 second(s) 
Memory limit: 262144 kilobytes

input: standard 
output: standard

ICPC (International Cardboard Producing Company) is in the business of producing cardboard boxes. Recently the company organized a contest for kids for the best design of a cardboard box and selected M winners. There are N prizes for the winners, each one carefully packed in a cardboard box (made by the ICPC, of course). The awarding process will be as follows:

• All the boxes with prizes will be stored in a separate room.
• The winners will enter the room, one at a time.
• Each winner selects one of the boxes.
• The selected box is opened by a representative of the organizing committee.
• If the box contains a prize, the winner takes it.
• If the box is empty (because the same box has already been selected by one or more previous winners), the winner will instead get a certificate printed on a sheet of excellent cardboard (made by ICPC, of course).
• Whether there is a prize or not, the box is re-sealed and returned to the room.

The management of the company would like to know how many prizes will be given by the above process. It is assumed that each winner picks a box at random and that all boxes are equally likely to be picked. Compute the mathematical expectation of the number of prizes given (the certificates are not counted as prizes, of course). 

Input

The first and only line of the input file contains the values of N and M (). 

Output

The first and only line of the output file should contain a single real number: the expected number of prizes given out. The answer is accepted as correct if either the absolute or the relative error is less than or equal to 10 ^-9. 

Example(s)

sample input	sample output
5 7	3.951424

sample input	sample output
4 3	2.3125

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InputThe first and only line of the input file contains the values of N and M ().OutputThe first and only line of the output file should contain a single real number: the expected number of prizes given out. The answer is accepted as correct if either the absolute or the relative error is less than or equal to 10 ^-9.Sample Input

sample input	sample output
5 7	3.951424

 

sample input	sample output
4 3	2.3125

 

 

题意：

有n个奖品，m个人排队来选礼物，对于每个人，他打开的盒子，可能有礼物，也有可能已经被之前的人取走了。为最后m个人取走礼物的期望。

思路：

排队取，第1个人取到1个，dp[1]=1;后面的人dp[i]=p取到礼物盒子+dp取到礼物=(n-dp[i-1])/n + dp[i-1]；

当然，也可以化简为公式   printf("%.10lf\n",n*1.0*(1-pow((n-1)*1.0/n,m)));  

#include<cstdio>
#include<cstdlib>
#include<iostream>
#include<algorithm>
#include<cstring>
#include<algorithm>
using namespace std;
double dp[100010];
int main()
{
    int n,m,i;
    while(~scanf("%d%d",&n,&m)){
        dp[1]=1;
        for(i=2;i<=m;i++) dp[i]=dp[i-1]+(n-dp[i-1])/n;
        printf("%.9lf\n",dp[m]);
    }return 0;
}