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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

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

class Main {

   

    // static final Reader in = new Reader();
    static final Scanner in = new Scanner(System.in);
    static final PrintWriter out = new PrintWriter(System.out);
    static int N, W;
    static final long MOD = 1000000007;
    static char[][] a;
    static double[][] dp;
    static double[] p;
    static double[] pre;


    static double P() {
        
        for(int i = 0;i < N;i++)
            dp[i][0] = 1;

        for(int i = 0;i < N;i++)
            dp[i][i+1] = pre[i];   
        

            for(int i = 1;i < N;i++){
                for(int k = 1;k < i+1;k++){
                    double consider = dp[i - 1][ k - 1];
                    double ignore = dp[i - 1][k];

                    dp[i][k] = (consider * p[i]) + (ignore * (1 - p[i]));
                }   
            }
        
        
        return dp[N - 1][N / 2 + 1];

    }

    public static void main(String[] args) throws Exception {
        N = in.nextInt();
        p = new double[N];
        pre = new double[N];
        dp = new double[N+1][N+1];

        
        p[0] = in.nextDouble();
        pre[0] = p[0];
        Arrays.fill(dp[0], 0);
        for (int i = 1; i < N; i++) {
            p[i] = in.nextDouble();
            pre[i] = pre[i - 1] * p[i];
            Arrays.fill(dp[i], 0);
        }

        System.out.println(P());

        out.close();

    }

}