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 PHP7 (7.0.15) (judge verdict: Accepted)

<?php
$n = (int) fgets(STDIN);
$probabilities = explode(" ", trim(fgets(STDIN)));
foreach ($probabilities as $k => $v) {
	$probabilities[$k] = (float) $v;
}
getHeadsSuperior($probabilities, $n);

function getHeadsSuperior($probabilities, $n) {
    $dp = [];
    for ($i = 0; $i <= $n; $i++){
        $dp[$i] = array_fill(0, $n + 1, 0.0);
    }
    $dp[0][0] = 1.0;
    for ($i = 1; $i < $n + 1; $i++) {
        for ($j = 0; $j < $n + 1; $j++) {
            $dp[$i][$j] = $dp[$i - 1][$j] * (1 - $probabilities[$i - 1]);
            if ($j > 0) {
                $dp[$i][$j] += $dp[$i - 1][$j - 1] * $probabilities[$i - 1];
            }
        }
    }
    $result = 0.0;
    for ($i = (int) ceil(($n + 1) / 2); $i <= $n; $i++) {
      $result += $dp[$n][$i];
    }
    echo $result;
}

?>