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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

# python3だとTLE,pypy3だとAC
import sys
sys.setrecursionlimit(10**6)


def get(i, j):
    global dp
    global p
    if i < 0 or j < 0:
        return 0.0
    if dp[i][j] == 0.0:
        dp[i][j] = get(i - 1, j) * (1 - p[i - 1]) + \
            get(i - 1, j - 1) * p[i - 1]
    return dp[i][j]


def main():
    N = int(input())
    global dp
    global p
    p = list(map(float, input().split()))
    dp = [[0.0] * (N + 1) for _ in range(N + 1)]
    dp[0][0] = 1.0
    for i in range(N, 0, -1):
        get(N, i)
    print(sum(dp[N][N//2+1:]))


if __name__ == "__main__":
    main()