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)

//package com.company;
// Always comment out package when submitting.

import java.io.*;
import java.util.*;

public class Main {
    // Actual Code
    public static class Task {


        public void solve(Scanner sc, PrintWriter pw) throws IOException {
            int n = sc.nextInt();
            double[] prob = new double[n];
            for (int i = 0; i < n; i++) {
                prob[i] = sc.nextDouble();
            }
            double[] dp = new double[n + 1];
            dp[0] = 1.0;
            for (int i = 0; i < n; i++) {
                for (int j = n; j >= 1; j--) {
                    dp[j] = dp[j - 1] * prob[i] + dp[j] * (1 - prob[i]);
                }
                dp[0] = dp[0] * (1 - prob[i]);
            }
            double ret = 0.0;
            for (int i = (n / 2) + 1; i <= n; i++) {
                ret += dp[i];
            }
            pw.println(ret);
        }
    }

    // template, actual code is in class Task.
    static long TIME_START, TIME_END;
    public static void main(String[] args) throws IOException {
        Scanner sc = new Scanner(System.in);
//        Scanner sc = new Scanner(new FileInputStream("File.in"));
        PrintWriter pw = new PrintWriter(new BufferedOutputStream(System.out));
//        PrintWriter pw = new PrintWriter(new FileOutputStream("File.out"));

        Runtime runtime = Runtime.getRuntime();
        long usedMemoryBefore = runtime.totalMemory() - runtime.freeMemory();
        TIME_START = System.currentTimeMillis();
        Task t = new Task();
        t.solve(sc, pw);
        TIME_END = System.currentTimeMillis();
        long usedMemoryAfter = runtime.totalMemory() - runtime.freeMemory();
        pw.close();
//        System.err.println("Memory increased:" + (usedMemoryAfter-usedMemoryBefore) / 1000000 );
//        System.err.println("Time used: " + (TIME_END - TIME_START) + ".");
    }

    // Faster IO with BufferedReader wrapped with Scanner
    static class Scanner {
        StringTokenizer st;
        BufferedReader br;

        public Scanner(InputStream s){br = new BufferedReader(new InputStreamReader(s));}

        public Scanner(FileReader s) throws FileNotFoundException {br = new BufferedReader(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();}
    }
}