lightOJ 1317 Throwing Balls into the Baskets

lightOJ  1317  Throwing Balls into the Baskets（期望）  解题报告

题目链接：http://acm.hust.edu.cn/vjudge/contest/view.action?cid=88890#problem/A

题目：

Description

You probably have played the game "Throwing Balls into the Basket". It is a simple game. You have to throw a ball into a basket from a certain distance. One day we (the AIUB ACMMER) were playing the game. But it was slightly different from the main game. In our game we were N people trying to throw balls into M identical Baskets. At each turn we all were selecting a basket and trying to throw a ball into it. After the game we saw exactly S balls were successful. Now you will be given the value of N and M. For each player probability of throwing a ball into any basket successfully is P. Assume that there are infinitely many balls and the probability of choosing a basket by any player is 1/M. If multiple people choose a common basket and throw their ball, you can assume that their balls will not conflict, and the probability remains same for getting inside a basket. You have to find the expected number of balls entered into the baskets after K turns.

Input

Input starts with an integer T (≤ 100), denoting the number of test cases.

Each case starts with a line containing three integers N (1 ≤ N ≤ 16), M (1 ≤ M ≤ 100) and K (0 ≤ K ≤ 100) and a real number P (0 ≤ P ≤ 1). P contains at most three places after the decimal point.

Output

For each case, print the case number and the expected number of balls. Errors less than 10^-6 will be ignored.

Sample Input

2

1 1 1 0.5

1 1 2 0.5

Sample Output

Case 1: 0.5

Case 2: 1.000000

题目大意：

有n个人，m个篮筐，一共打了k轮，每轮每个人可以投一个球，每个球投进的概率都是p,求k轮后，投中的球的期望是多少？

分析：

因为每个人投进的概率都是相同的，所以期望也是相同的。因此只需要求出第一轮的期望就可以了，总期望=k*第一轮的期望。

代码：

#include<cstdio>
#include<iostream>
using namespace std;

int t,n,m,k;
double p,ans;
int a[20][20];

void init()
{
    a[1][1]=1;
    a[1][0]=1;
    for(int i=2;i<20;i++)
    {
        a[i][i]=1;
        a[i][0]=1;
        for(int j=1;j<i;j++)
            a[i][j]=a[i-1][j]+a[i-1][j-1];
    }
}

double count(int j)
{
    double b=1.0;
    for(int i=0;i<j;i++)
        b=b*p;//投中的期望
    for(int i=0;i<n-j;i++)
        b=b*(1.0-p);//没投中的期望
    return b*j*a[n][j];
}

int main()
{
    int c=1;
    scanf("%d",&t);
    init();
    while(t--)
    {
        scanf("%d%d%d%lf",&n,&m,&k,&p);
        ans=0.0;//小数
        for(int i=0;i<=n;i++)
            ans+=count(i);//第一轮的期望
        printf("Case %d: %.7lf
",c++,ans*k);
    }
    return 0;
}