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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
#define pii pair<int,int>
#define sorted(a_1) sort(a_1.begin(),a_1.end())
#define rsorted(a_1) sort(a_1.rbegin(),a_1.rend())
#define t1(a_1) cout<<a_1<<endl;
#define t2(a_1) for(auto it_test : a_1) cout<<it_test<<" ";
#define MOD 1000000007

long double dp[3001][3001];
long double probability(vector<double>&arr, int index, int tail){
	if(index<0 || tail<0){ return 1.0;}
	if(dp[tail][index]>0.0) return dp[tail][index];
	if(tail==0){
		dp[tail][index]=arr[index]*probability(arr,index-1,tail);
		return dp[tail][index];
	}
	else{
		dp[tail][index]=(1-arr[index])*probability(arr,index-1,tail-1)+arr[index]*probability(arr,index-1,tail);
		return dp[tail][index];
	}
}
void solve(){
    int n;
    cin>>n;
    vector<double>arr(n);
    for(int i=0;i<n;i++){
    	cin>>arr[i];
    }
    memset(dp,0,sizeof(dp));
  	long double ans = probability(arr,n-1,n/2);
  	cout<<ans;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int t=1;
    while(t--){
        solve();
    }

return 0;
}