uva 11021 数学概率 麻球

GRAVITATION, n.
“The tendency of all bodies to approach one another with a strength
proportion to the quantity of matter they contain – the quantity of
matter they contain being ascertained by the strength of their tendency
to approach one another. This is a lovely and edifying illustration of
how science, having made A the proof of B, makes B the proof of A.”
Ambrose Bierce
You have a population of k Tribbles. This particular species of Tribbles live for exactly one day and
then die. Just before death, a single Tribble has the probability Pi of giving birth to i more Tribbles.
What is the probability that after m generations, every Tribble will be dead?
Input
The first line of input gives the number of cases, N. N test cases follow. Each one starts with a line
containing n (1 ≤ n ≤ 1000), k (0 ≤ k ≤ 1000) and m (0 ≤ m ≤ 1000). The next n lines will give the
probabilities P0, P1, … , Pn−1.
Output
For each test case, output one line containing ‘Case #x:’ followed by the answer, correct up to an
absolute or relative error of 10−6
.
Sample Input
4
3 1 1
0.33
0.34
0.33
3 1 2
0.33
0.34
0.33
3 1 2
0.5
0.0
0.5
4 2 2
0.5
0.0
0.0
0.5
Sample Output
Case #1: 0.3300000
Case #2: 0.4781370
Case #3: 0.6250000
Case #4: 0.3164062

题意：k只麻球，每活一天就会死亡，但第二天可能会生一些麻球，具体是 生i个麻球的概率为pi ,求m天后所有麻球都死亡的概率。

思路：考虑全概率公式，求k只麻球m天后全死亡 ，因为死亡是独立事件，应用乘法 ，ans= f[m] ^k ,f[m] 为一只麻球m天后均死亡的概率。对于第i天，

f[i]=p0+p1*f[i-1]^1 +p2*f[i-1]^2 +…p(n-1)*f[i-1]^(n-1) （就是让i-1天所有的出生的麻球全部死亡，那么第i天麻球就没了。。）

f[m] 的概率有 f【m-1】得出。对，就是让第i-1天出生的麻球全部死亡。生产麻球的数目之间相加即可。

#include <bits/stdc++.h>using namespace std;double f[1010];double p[1010];int main(){    int t;    cin>>t;    int cas=1;    while(t--)    {        int n,m,k;        cin>>n>>k>>m;        for(int i=0;i<n;i++)            scanf("%lf",&p[i]);        f[0]=0.0;        f[1]=p[0];        for(int i=2;i<=m;i++)          {              f[i]=0;              for(int j=0;j<n;j++)              {                  f[i]+=p[j]*pow(f[i-1],j);              }          }          double res=pow(f[m],k);        printf("Case #%d: %.7f\n",cas++,res);    }}