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 JavaScript (Node.js 12.16.1) (judge verdict: Accepted)


function Main(input) {
	input = input.trim().split("\n").map(function(x) { return x.split(" ")});    
	let N = parseInt(input[0][0], 10);
	let p = input[1].map(e => parseFloat(e, 10));
	let maxUra = Math.ceil((N + 1) / 2);
	let DP = Array.from({length: maxUra}, () => 0);
	DP[0] = 1;
	for (let i = 0; i < N; i++){
		for (let j = maxUra - 1; j >= 1; j--){
			DP[j] = DP[j - 1] * (1 - p[i]) + DP[j] * p[i];
		}
		DP[0] = DP[0] * p[i];
		// console.log(DP);
	}
	let ans = 0;
	for (x of DP){
		ans += x;
	}
	console.log(ans);
}

Main(require("fs").readFileSync("/dev/stdin", "utf8"));