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 JavaScript (Node.js 12.16.1) (judge verdict: Wrong Answer)

'use strict';

process.stdin.resume();
process.stdin.setEncoding('utf-8');

let inputString = '';
let currentLine = 0;

process.stdin.on('data', inputStdin => {
    inputString += inputStdin;
});

process.stdin.on('end', _ => {
    inputString = inputString.trim().split('\n').map(string => {
        return string.trim();
    });

    main();
});

function readline() {
    return inputString[currentLine++];
}

function getArray(str) {
    return str.split(' ').map(x => Number(x));
}

function getInt(str) {
    return Number(str);
}

function getStrArray(str) {
    return str.split(' ').map(x => x);
}

function create2DArray(row, col, filler = 0) {
    return Array(row)
        .fill(filler)
        .map(() => Array(col).fill(filler));
}
// ---------------------------------------



function main() {
    let n = getInt(readline());
    let ar = getArray(readline());
    let arr = [0, ...ar];
    let dp = create2DArray(n+1, n+1, 0);
    dp[1][0] = 1-arr[1];
    dp[1][1] = arr[1];
    for (let i = 2; i <= n; ++i) {
        dp[i][0] = dp[i][0]* (1-arr[i]);
    }

    for (let i = 2; i <= n; ++i) {
        dp[i][i] = dp[i - 1][i - 1] * arr[i];
    }
    

    for(let i=1; i <= n; ++i) {
        for (let j = 1; j < i; ++j) {
            dp[i][j] = dp[i-1][j-1]*arr[i] + dp[i-1][j]*(1-arr[i])
        }    
    }

    let ans = 0;

    for (let i = Math.ceil(n/2); i <= n; ++i) {
        ans +=dp[n][i];
    }
    console.log(ans);
} 

/*

dp[i][j] = dp[i-1][j-1]*pj +  dp[i-1][j]*(1-pj)








*/