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light oj 1248 - Dice (III)（期望）

来源：程序员人生发布时间：2015-05-20 10:47:56 阅读次数：3517次

Given a dice with n sides, you have to find the expected number of times you have to throw that dice to see all its faces at least once. Assume that the dice is fair, that means when you throw the dice, the probability of occurring any face is equal.

For example, for a fair two sided coin, the result is 3. Because when you first throw the coin, you will definitely see a new face. If you throw the coin again, the chance of getting the opposite side is 0.5, and the chance of getting the same side is 0.5. So, the result is

1 + (1 + 0.5 * (1 + 0.5 * ...))

= 2 + 0.5 + 0.5² + 0.5³ + ...

= 2 + 1 = 3

Input

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

Each case starts with a line containing an integer n (1 ≤ n ≤ 10⁵).

Output

For each case, print the case number and the expected number of times you have to throw the dice to see all its faces at least once. Errors less than 10^⑹ will be ignored.

Sample Input

Output for Sample Input

100

Case 1: 1

Case 2: 3

Case 3: 5.5

Case 4: 14.7

Case 5: 518.7377517640

#include<iostream> #include<cstdio> #include<cstring> #include<algorithm> #include<cmath> #include<queue> #include<stack> #include<vector> #include<set> #include<map> #define L(x) (x<<1) #define R(x) (x<<1|1) #define MID(x,y) ((x+y)>>1) #define eps 1e⑻ //typedef __int64 ll; #define fre(i,a,b) for(i = a; i <b; i++) #define free(i,b,a) for(i = b; i >= a;i--) #define mem(t, v) memset ((t) , v, sizeof(t)) #define ssf(n) scanf("%s", n) #define sf(n) scanf("%d", &n) #define sff(a,b) scanf("%d %d", &a, &b) #define sfff(a,b,c) scanf("%d %d %d", &a, &b, &c) #define pf printf #define bug pf("Hi ") using namespace std; #define INF 0x3f3f3f3f #define N 100005 double dp[N]; int n; int main() { int i,j,t,ca=0; sf(t); while(t--) { sf(n); mem(dp,0); for(i=n⑴;i>=0;i--) dp[i]=((n-i)*dp[i+1]+n)*1.0/(n-i); pf("Case %d: %.7lf ",++ca,dp[0]); } return 0; }

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