Problem:
Score : 100 points

Problem Statement

Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N.
For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins.
Find the probability of having more heads than tails.

Constraints

N is an odd number.

1 \leq N \leq 2999

p_i is a real number and has two decimal places.

0 < p_i < 1

Input

Input is given from Standard Input in the following format:

N
p_1 p_2 \ldots p_N

Output

Print the probability of having more heads than tails.
The output is considered correct when the absolute error is not greater than 10^{-9}.

Sample Input 1

3
0.30 0.60 0.80

Sample Output 1

0.612

The probability of each case where we have more heads than tails is as follows:

The probability of having (Coin 1, Coin 2, Coin 3) = (Head, Head, Head) is 0.3 × 0.6 × 0.8 = 0.144;

The probability of having (Coin 1, Coin 2, Coin 3) = (Tail, Head, Head) is 0.7 × 0.6 × 0.8 = 0.336;

The probability of having (Coin 1, Coin 2, Coin 3) = (Head, Tail, Head) is 0.3 × 0.4 × 0.8 = 0.096;

The probability of having (Coin 1, Coin 2, Coin 3) = (Head, Head, Tail) is 0.3 × 0.6 × 0.2 = 0.036.

Thus, the probability of having more heads than tails is 0.144 + 0.336 + 0.096 + 0.036 = 0.612.

Sample Input 2

1
0.50

Sample Output 2

0.5

Outputs such as 0.500, 0.500000001 and 0.499999999 are also considered correct.

Sample Input 3

5
0.42 0.01 0.42 0.99 0.42

Sample Output 3

0.3821815872

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#define _USE_MATH_DEFIMES
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <climits>
#include <clocale>
#include <cmath>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <ctime>
#include <deque>
#include <fstream>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <regex>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

const int MOD = 1'000'000'007;
const int INF = 1'000'000'000; //1e9
const int NIL = -1;
const long long LINF = 1'000'000'000'000'000'000; // 1e18
const long double EPS = 1E-10;

template<class T, class S> inline bool chmax(T &a, const S &b){
    if(a < b){
        a = b; return true;
    }
    return false;
}
template<class T, class S> inline bool chmin(T &a, const S &b){
    if(b < a){
        a = b; return true;
    }
    return false;
}








int main(){
    int N; std::cin >> N;
    std::vector<long double> p(N);
    for(int i(0); i < N; ++i) std::cin >> p[i];
    // i枚投げてj枚表
    std::vector<std::vector<long double>> dp(N+1, std::vector<long double>(N+1, 0.0L));
    dp[1][0] = 1 - p[0]; dp[1][1] = p[0];
    for(int i(2); i <= N; ++i){
        for(int j(0); j <= i; ++j){
            if(!j) dp[i][j] = dp[i-1][j] * (1 - p[i-1]);
            else dp[i][j] = dp[i-1][j-1] * p[i-1] + dp[i-1][j] * (1 - p[i-1]);
        }
    }
    long double ans(0.0L);
    for(int i(N/2 + 1); i <= N; ++i){
        ans += dp[N][i];
    }
    std::cout << std::fixed << std::setprecision(10) << ans << std::endl;
    return 0;
}