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: Wrong Answer)

#include<bits/stdc++.h>
#define int long long
#define double long double
#define vi vector<int>
#define vpi vector<pair<int,int> >
#define mii map<int,int>
#define mpi map<pair<int,int>,int>
#define msi map<string,int>
#define set set<int>
#define u_set unordered_set<int>
#define pb push_back
#define endl "\n"
#define ll long long int
#define vll vector<long long int>
#define vi vector<int>
#define vvi vector<vector<int>>
#define io ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
#define umap unordered_map 
#define mod 1000000007
#define fori(a,n) for(int i=a;i<n;i++)
#define forj(a,n) for(int j=a;j<n;j++)
#define MAX 300303
#define V vector
using namespace std;
signed main()
{
    io
    int n;
    cin >> n;
    V<double> p(n);
    fori(0,n)cin >> p[i];
    V<V<double>> dp(n+5,V<double>(n+5,0.0));
    dp[0][0] = 1-p[0];
    dp[0][1] = p[0];
    fori(1,n){
        forj(0,i+2){
            if(j>0)
                dp[i][j] = dp[i-1][j-1]*p[i] + dp[i-1][j]*(1-p[i]);
            else dp[i][j] = dp[i-1][j]*(1-p[i]);
        }
    }
    double ans = 0 ;
    forj(n/2+1,n+1){
            ans += dp[n-1][j];
    }
    cout << ans << endl;
    
}