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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE RankNTypes #-}
import qualified Data.ByteString.Lazy.Char8 as B
import Data.Maybe
import Data.List
import qualified Data.Vector.Unboxed as VU
-- import qualified Data.Vector.Unboxed.Mutable as VUM
-- import Control.Monad
-- import Control.Monad.ST
-- import Debug.Trace
-- trace _ = id

solve :: Int -> [Double] -> Double
solve n ps = pw
  where
    res = foldl' op (VU.fromList [1]) ps
    op vec p = VU.zipWith (+)
               (VU.snoc (VU.map (* p) vec) 0)
               (VU.cons 0 (VU.map (* (1-p)) vec))
    pw = VU.sum (VU.take ((n+1) `div` 2) res)

readBInt :: B.ByteString -> Int
readBInt = fst . fromJust . B.readInt

tmain :: B.ByteString -> Double
tmain cont =
  let remLines0 = map B.words (B.lines cont)
      [bs_n]:remLines1 = remLines0
      n = readBInt bs_n
      line2:remLines2 = remLines1
      ps = map (read . B.unpack) line2
  in solve n ps

outAnswer :: Double -> IO ()
outAnswer x = putStrLn (show x)

main :: IO ()
main = outAnswer . tmain =<< B.getContents