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 IOS ios_base::sync_with_stdio(false); cin.tie(NULL)
#define ll long long

using namespace std;

double maxHeadProbab(vector<double> &p,int cur,int tc, vector<vector<double>> &dp){

    if(cur == p.size()){
        return 1;
    }

    if(dp[cur][tc] != -1){
        return dp[cur][tc];
    }

    double t = 0, h = 0;
    if (tc < p.size() / 2)
    {
        t = (1 - p[cur]) * maxHeadProbab(p, cur + 1,tc + 1,dp);
    }
    h = p[cur] * maxHeadProbab(p, cur + 1,tc,dp);
    dp[cur][tc] = h + t;
    return h + t;
}

void solve()
{
    int n;
    cin >> n;

    vector<double> pro;
    vector<vector<double>> dp(n + 1, vector <double>((n + 1 / 2), -1));
    for (int i = 0; i < n; i++)
    {
        double prob1;
        cin >> prob1;
        pro.push_back(prob1);
    }

    cout << maxHeadProbab(pro, 0, 0, dp);
}
int main()
{
    IOS;
    // int t;
    // cin>>t;
    // while(t>0)
    // {
        solve();
    //     t--;
    // }
    return 0;
}