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: Time Limit Exceeded)

#include<bits/stdc++.h>
using namespace std;
typedef long long int ll;
#define fi first
#define sec second 
#define MOD 1000000007
#define pb push_back
#define endl "\n"
#define test cout<<"debuged\n";
#define INF 10000000000
#define si 1000000
#define pi pair<ll,ll>
#define debug(x) cout<<x<<"\n";

vector<double> v(si,0);
vector<double> ans;

double dp[3010][3010];

double solve(ll i,ll n,ll c)
{
  if(c<=0 and i==n)
  {
    return 1;
  }
  if(i>=n)
    return 0;
  if(dp[i][c]!=0)
    return dp[i][c];
  double a=solve(i+1,n,c-1);
  double b=solve(i+1,n,c);
  a=a*v[i];
  b=b*(1-v[i]);
  dp[i][c]=a+b;
  return a+b;
} 

int main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
   
    ll n;
    cin>>n;
    for(ll i=0;i<n;i++) 
      cin>>v[i];
    ll c=n/2;
    if(n%2==1)
      c++;
    double val=1.0,sum=0;
    cout<<fixed;
    for(ll i=0;i<n+1;i++)
      for(ll j=0;j<n+1;j++)
        dp[i][j]=0;
    cout<<setprecision(9);
    cout<<solve(0,n,c);  
}