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.util.StringTokenizer;

public class Main {

    public static void main(String[] args) throws IOException {
        init();

        int N = nextInt();

        double[] probability = new double[N];

        for (int i = 0; i < probability.length; i++) probability[i] = nextDouble();

        double[][] coinProbabilities = new double[N + 1][N + 1];

        coinProbabilities[0][0] = 1;

        for (int i = 0; i < coinProbabilities.length; i++) {
            for (int j = 0; i + j <= N; j++) {
                int numOfCoins;
                if (i > 0) {
                    numOfCoins = j + i - 1;
                    coinProbabilities[i][j] += coinProbabilities[i - 1][j] * probability[numOfCoins];
                }

                if (j > 0) {
                    numOfCoins = i + j - 1;
                    coinProbabilities[i][j] += coinProbabilities[i][j - 1] * (1 - probability[numOfCoins]);
                }
            }
        }

        double ans = 0.0;
        for (int i = 0; i < coinProbabilities.length; i++) {
            for (int j = 0; j < coinProbabilities.length; j++) {
                if (i + j == N && i > j) ans += coinProbabilities[i][j];
            }
        }

        System.out.println(ans);
    }

    //Input Reader
    private static BufferedReader reader;
    private static StringTokenizer tokenizer;

    private static void init() {
        reader = new BufferedReader(new InputStreamReader(System.in));
        tokenizer = new StringTokenizer("");
    }

    private static String next() throws IOException {
        String read;
        while (!tokenizer.hasMoreTokens()) {
            read = reader.readLine();
            if (read == null || read.equals(""))
                return "-1";
            tokenizer = new StringTokenizer(read);
        }

        return tokenizer.nextToken();
    }

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

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