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: Runtime Error)

#include<bits/stdc++.h>
using namespace std;
#define ll long long int
#define pb push_back
#define mp make_pair
#define nt _int128
#define mod 1000000007
double arr[3005],dp[3005][3005];
ll n;
double func(ll i,ll k)
{
    if(i==n+1)
    {
        if(k==0)
            return 1.0;
        else 
            return 0.0;
    }
    else if(dp[i][k]!=-1.0)
    {
        return dp[i][k];
    }

    double x=0;
    if(k>0)
        x+=arr[i]*func(i+1,k-1);
    x+=(1-arr[i])*func(i+1,k);
    return dp[i][k]=x;
}
int main()
{
#ifndef ONLINE_JUDGE
       freopen("input.txt","r",stdin);
    freopen("output.txt","w",stdout);
#endif
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    ll t;
    // cin >> t;
    t  = 1;
    while(t--)
    {
        cin>>n;
        for(ll i=1;i<=n;i++)
            cin>>arr[i];
        double x=0;
        for(ll i=1;i<=3000;i++)
        {
            for(ll j=0;j<=3000;j++)
                dp[i][j]=-1.0;
        }
        for(ll i=n/2+1;i<=n;i++)
        {
            x+=func(1,i);
        }
        // for(ll i=1;i<=n;i++)
        // {
        //     for(ll j=0;j<=2;j++)
        //         cout<<dp[i][j]<<" ";
        //     cout<<endl;
        // }
        cout<<fixed<<setprecision(10)<<x;
    }
    return 0;
}