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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.lang.reflect.Array;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Stack;
import java.util.StringTokenizer;
public class Main{
    static int n;
    static double[] prob;
    static double[][] memo;
    public static void main(String[] args) throws IOException {
        scanner sc = new scanner(System.in);
        PrintWriter out = new PrintWriter(System.out);

        n = sc.nextInt();

        prob = new double[n];
        for (int i = 0; i < n; i++)
            prob[i] = sc.nextDouble();

        memo = new double[n][n+1];

        for (int i = 0; i < n; i++)
            Arrays.fill(memo[i],-1);





         out.println(dp(0,0));
        out.flush();

    }

    static double dp(int i,int heads){
        if (i == n)
            return heads > n/2?1:0;

        if (memo[i][heads] != -1) return memo[i][heads];

        return memo[i][heads] = prob[i]*dp(i+1,heads+1) + (1-prob[i])*dp(i+1,heads);
    }

}
class scanner{
    StringTokenizer st;
    BufferedReader br;
    public scanner(InputStream s){
        br = new BufferedReader(new InputStreamReader(s));
    }
    public scanner (String s) throws FileNotFoundException { br = new BufferedReader(new FileReader(s));}

    public String next() throws IOException {
        while (st == null || !st.hasMoreTokens())
            st = new StringTokenizer(br.readLine());
        return st.nextToken();
    }

    public int nextInt() throws IOException {
        return Integer.parseInt(next());
    }

    public long nextLong() throws IOException {
        return  Long.parseLong(next());
    }

    public String nextLine() throws IOException {
        return br.readLine();
    }

    public double nextDouble() throws IOException {
        return Double.parseDouble(next());
    }

    public boolean ready() throws IOException {
        return br.ready();
    }
}