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 Scala (2.11.7) (judge verdict: Time Limit Exceeded)

object Main extends App {

  import scala.io.StdIn._

  val n = readLine.trim.toInt
  val probs = readLine.trim.split(' ').map(_.toDouble)
  println(cal(probs))
  def cal(prob: Array[Double]): Double = {
    val memo = Array.tabulate(prob.length + 1){rest ⇒ Array.tabulate(rest + 1){_ ⇒ None:Option[Double]}}
    def sub(rest: Int, sur: Int): Double = {
      if (rest < sur) 0d
      else if (rest == 0) 1d
      else if (sur == 0) 1d
      else {
        memo(rest)(sur) match {
          case Some(r) ⇒ r
          case None ⇒
            (prob(rest - 1) * sub(rest - 1, sur - 1) + (1 - prob(rest - 1)) * sub(rest - 1, sur)).also{r ⇒ memo(rest)(sur) = Some(r)}
        }
      }
    }
    for (i ← prob.indices by 10) {
      for (j ← 0 until i by 10){
        sub(i, j)
      }
    }
    sub(prob.length, prob.length / 2 + 1)
  }
  implicit class Extend[T](value: T){
    def also(func: T ⇒ Unit): T = {func(value); value}
    def let[A](func: T ⇒ A): A = func(value)
  }
}