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.Arrays;
import java.util.Scanner;

public class Main {
    static final Scanner sc  = new Scanner(System.in);
    static final int MOD = (int) 1E9 + 7;
    static final int INF = Integer.MAX_VALUE;
    
    public static void main(String[] args) {
        int N = nint();
        double[] p = ndoubles(N, 1, 0);
        int K = N*2 + 1;
        double[][] dp = new double[K+2][N+1];
        dp[N+1][0] = 1;
        
        for (int n = 1; n <= N; n++) {
            for (int k = 1; k <= K; k++) {
                dp[k][n] += dp[k-1][n-1] * p[n];
                dp[k][n] += dp[k+1][n-1] * (1 - p[n]);
            }
        }
        
        double ans = 0;
        for (int k = N+1; k <= K; k++) {
            ans += dp[k][N];
        }
        System.out.println(ans);
    }
    
    
    static long max(long... longs) {
        return Arrays.stream(longs).max().getAsLong();
    }
    
    static long min(long... longs) {
        return Arrays.stream(longs).min().getAsLong();
    }

    private static long nlong() {
        return sc.nextLong();
    }

    private static int nint() {
        return sc.nextInt();
    }

    private static String nstr() {
        return sc.next();
    }

    private static char[] nchars() {
        return sc.next().toCharArray();
    }

    private static char[] nchars(int padL, int padR) {
        char[] temp = sc.next().toCharArray();
        char[] ret = new char[temp.length + padL + padR];
        System.arraycopy(temp, 0, ret, padL, temp.length);
        return ret;
    }

    private static int[] nints(int n) {
        return nints(n, 0, 0);
    }

    private static int[] nints(int n, int padL, int padR) {
        int[] a = new int[padL + n + padR];
        for (int i = 0; i < n; i++)
            a[padL + i] = nint();
        return a;
    }
    
    private static double[] ndoubles(int n) {
        return ndoubles(n, 0, 0);
    }
    
    private static double[] ndoubles(int n, int padL, int padR) {
        double[] d = new double[n + padL + padR];
        for (int i = 0; i < n; i++) {
            d[padL + i] = sc.nextDouble();
        }
        return d;
    }

    private static char[][] nchars2(int h, int w) {
        return nchars2(h, w, 0);
    }

    private static char[][] nchars2(int h, int w, int pad) {
        char[][] a2 = new char[h + pad * 2][w + pad * 2];
        for (int i = 0; i < h; i++)
            System.arraycopy(nchars(), 0, a2[pad + i], pad, w);
        return a2;
    }
}