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.Scanner;
import java.util.Arrays;

public class Main {
    private static Scanner in;

    private static int n;
    private static double[] p;
    private static double[][] dp;

	public static void main(String[] args) {
        in = new Scanner(System.in);
        
        n = in.nextInt();
        p = new double[n];
        dp = new double[n][n];
        for(int i=0; i<n; i++) {
            Arrays.fill(dp[i], -1);
        }

        for(int i=0; i<n; i++) p[i] = in.nextDouble();

        System.out.println(solve(0, 0));
	}

    private static double solve(int i, int heads) {
        if (i==n) {
            if (heads>n-heads) {
                return 1;
            } else {
                return 0;
            }
        }

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

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

}