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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)

import java.util.*;

public class Main{
	
	private static Scanner sc=new Scanner(System.in);
	private static final int MOD=1000000007;
	
		
	public static double printComb(int coins,int H,int T,double []prob,double hprob){
		if(coins==0){
			if(H>T){
				return hprob;
			}
			else{
				return 0;
			}
		}
		double ans=0.0;
		ans+=printComb(coins-1,H+1,T,prob,hprob*prob[coins-1]);
		ans+=printComb(coins-1,H,T+1,prob,hprob*(1-prob[coins-1]));
		return ans;
	}
	
	
	public static void main(String []args){
		int n=sc.nextInt();
		double hp=1.0;
		double tp=1.0;
		double prob[]=new double[n];
		for(int i=0;i<n;i++){
			prob[i]=sc.nextDouble();
		}
		System.out.println(printComb(n,0,0,prob,1));
	}
	
}