【light-oj】-1104 - Birthday Paradox（数学，概率）

1104 - Birthday Paradox

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Time Limit: 2 second(s)Memory Limit: 32 MB

Sometimes some mathematical results are hard to believe. One of the common problems is the birthday paradox. Suppose you are in a party where there are 23 people including you. What is the probability that at least two people in the party have same birthday? Surprisingly the result is more than 0.5. Now here you have to do the opposite. You have given the number of days in a year. Remember that you can be in a different planet, for example, in Mars, a year is 669 days long. You have to find the minimum number of people you have to invite in a party such that the probability of at least two people in the party have same birthday is at least 0.5.

Input

Input starts with an integer T (≤ 20000), denoting the number of test cases.

Each case contains an integer n (1 ≤ n ≤ 10⁵) in a single line, denoting the number of days in a year in the planet.

Output

For each case, print the case number and the desired result.

Sample Input

Output for Sample Input

2

365

669

Case 1: 22

Case 2: 30

题意：我们知道有一个有关于生日的结论：就是如果有23个人同时在一个教室，则这个教室至少有2个人同时生日的概率>50% .  给出n，表示一年有n天，求一个人至少邀请多少个人，才可以至少有2个人同时生日的概率>=0.5

求解：

设教室有i个人，但是没有人同时生日的概率为 p(i)

则：p(1)=1

i>1时：第一个人有365种选择，第2个人只有364种选择，第3个人有363种选择...

则：p(i)=(365/365)*(364/365)*(363/365)*...*((365-i+1)/365)

        =(265*364*...*(366-i))/(365^i)

则当p(i)<=0.5时，至少2个人同时生日的概率为1-p(i)>=0.5

所以我们只需要知道没有人同时生日的概率<=0.5时人数 i，就知道了至少两人同时生日的概率>=0.5的人数.

注意：

如果一年只有1天的话，需要邀请1个人。

#include<cstdio>#include<cstring>#include<algorithm>using namespace std;#define INF 0x3f3f3f3f#define CLR(a,b) memset(a,b,sizeof(a))#define LL long longint main(){int u,n,i,ca=1;scanf("%d",&u);while(u--){i=1;scanf("%d",&n);double p=1;//概率 for(;i<n;i++){p=p*(n-i)/n;if(p<=0.5)//概率<=0.5,i个人至少两人同时生日概率就>=0.5 break;}if(n==1)printf("Case %d: 1\n",ca++);elseprintf("Case %d: %d\n",ca++,i);} return 0;}