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>
using namespace std;
typedef long long ll;
#define MOD 1e9 + 7

template < typename T >
void printVector (const vector < T > & vec) {
	for (const T x : vec) {
		cout << x << " ";
	}
	cout << endl;
}

template < typename T >
void printMatrix (const vector < vector < T > > & mat) {
	for (const vector < T > vec : mat) {
		printVector (vec);
	}
	cout << endl;
}

const ll INF = 1e18L + 5;

template < typename T >
void min_self (T& a, T b) {
	a = min (a, b);
}

template < typename T >
void max_self (T& a, T b) {
	a = max (a, b);
}

void add_self (int & a, int b) {
	a += b;
	if (a >= MOD) {
		a -= MOD;
	}
}

int main() {
	int n;
	double p;
	scanf ("%d", &n);
	vector < double > dp (n + 1, 0.0);
	dp[0] = 1.0;
	for (int coin = 1; coin <= n; coin++) {
		scanf ("%lf", &p);
		for (int i = coin; i >= 0; i--) {
			dp[i] = (i == 0 ? 0 : dp[i - 1] * p) + dp[i] * (1.0 - p);
		}
	}
	double answer = 0.0;
	for (int i = n / 2 + 1; i <= n; i++) {
		answer += dp[i];
	}
	printf ("%lf\n", answer);
	return 0;
}