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: Accepted)

#include <iostream>
#include <vector>
#include <string>
#include <cmath>
#include <algorithm>
#include <map>
#include <iomanip> // for /* << std::fixed << std::setprecision(k) << */
using namespace std;

//defines

#define forn(n) for(int i = 0; i < n; i++)
#define pb push_back
#define int long long

int32_t main() {
	int n;
	
	cin >> n;
	
	vector<double> p(n);
	
	for (int i = 0; i < n; i++)
		cin >> p[i];
		
	vector<vector<double>> dp(n+1, vector<double>(n+1));
	
	dp[0][0] = 1;
	
	for (int i = 1; i <= n;i++)
		dp[0][i] = 0;
	
	for (int i = 1; i <= n;i++) {
		dp[i][0] = dp[i-1][0]*(1-p[i-1]);
		
		for (int j = 1; j <= n; j++) 
			dp[i][j] = dp[i-1][j]*(1-p[i-1]) + dp[i-1][j-1]*p[i-1];
	}
		
	double ans = 0;
	
	for (int i = (n+1)/2;i <= n;i++)
		ans += dp[n][i];
		
	cout << std::fixed << std::setprecision(15) << ans;
}