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.

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}.

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

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

见题面 Input 部分

见题面 Output 部分