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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::{input, fastout};

#[fastout]
fn main() {
    input! {
        n: usize,
        p: [f64; n],
    }

    // dp[i][j] := i枚目まで投げて表がj枚出る確率
    // dp[i+1][j] = dp[i][j-1] * p[i] + dp[i][j] * p[i-1];
    let mut dp: Vec<Vec<f64>> = vec![vec![0.0; n+1]; n+1];
    dp[1][1] = p[0];
    dp[1][0] = 1.0 - p[0];

    for i in 1..n {
        dp[i+1][0] = dp[i][0] * (1.0 - p[i]);
        for j in 1..=n {
            dp[i+1][j] = dp[i][j-1] * p[i] + dp[i][j] * (1.0 - p[i]);
        }
    }

    let mut ans: f64 = 0.0;
    for i in 0..=n {
        if i > n/2 {
            ans += dp[n][i];
        }
    }
    println!("{:.10}", ans);
}