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.util.*;
import java.io.*;
public class Main{
    static int N;
    static double dp[][], p[], ans=0.0;
    public static void main(String[] args) throws IOException{
        N = readInt();
        dp = new double[N+1][N+1];
        p = new double[N+1];
        for (int i = 1; i<=N; i++) p[i] = readDouble();
        dp[1][1] = p[1];
        dp[1][0] = 1.0 - p[1];
        for (int i = 2; i<=N; i++){
            dp[i][0] = dp[i-1][0]*(1.0-p[i]);
            for (int j = 1; j<=i; j++){
                dp[i][j] = dp[i-1][j-1]*p[i] + dp[i-1][j]*(1.0-p[i]);
            }
        }
        for (int i = N/2 + 1; i<=N; i++) ans+=dp[N][i];
        System.out.println(ans);
    }
    static BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
    static StringTokenizer st;
    static String next() throws IOException {
            while (st == null || !st.hasMoreTokens())
                    st = new StringTokenizer(br.readLine().trim());
            return st.nextToken();
    }
    static long readLong() throws IOException {
            return Long.parseLong(next());
    }

    static float readFloat() throws IOException {
            return Float.parseFloat(next());
    }

    static boolean readBool() throws IOException {
            return Boolean.parseBoolean(next());
    }

    static short readShort() throws IOException {
            return Short.parseShort(next());
    }

    static byte readByte() throws IOException {
            return Byte.parseByte(next());
    }

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

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

    static char readChar() throws IOException {
            return next().charAt(0);
    }

    static String readLine() throws IOException {
            return br.readLine().trim();
    }

    static PrintWriter pr = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));

    static void print(Object o) {
            pr.print(o);
    }
}