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 (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
use std::io::{ self, Write };
use std::str::FromStr;
use std::cmp::{ min, max };
use std::collections::{ BinaryHeap, VecDeque };

#[allow(unused_macros)]
macro_rules! trace {
    ($var:expr) => {
        let _ = writeln!(&mut std::io::stderr(), ">>> {} = {:?}", stringify!($var), $var);
    };
    ($($args:expr),*) => { trace!(($($args),*)) }
}

#[allow(unused_macros)]
macro_rules! put {
    ($var:expr) => {
        let _ = writeln!(&mut std::io::stdout(), "{}", $var);
    };
    ($var:expr, $($args:expr),*) => {
        let _ = write!(&mut std::io::stdout(), "{} ", $var);
        put!($($args),*);
    };
}

fn main() {
    let mut sc = Scanner::new();
    let n: usize = sc.cin();
    let ps = (0..n).map(|_| sc.cin::<f64>()).collect::<Vec<_>>();

    let mut p = vec![vec![0.0f64; 3000]; 3000];  // p[i][j] = Pr(j coins at i coins)
    p[0][0] = 1.0;

    for m in 1..n+1 {
        for k in 0..m+1 {
            p[m][k] = if k == 0 { 0.0 } else { p[m-1][k-1] * ps[m-1] } + p[m-1][k] * (1.0 - ps[m-1]);
            // trace!(m, k, p[m][k]);
        }
    }

    let ans: f64 = (n/2+1..n+1).map(|i| p[n][i]).sum();
    put!(ans);
}

#[allow(dead_code)]
struct Scanner { stdin: io::Stdin, buffer: VecDeque<String>, }
#[allow(dead_code)]
impl Scanner {
    fn new() -> Scanner { Scanner { stdin: io::stdin(), buffer: VecDeque::new() } }
    fn reserve(&mut self) {
        while self.buffer.len() == 0 {
            let mut line = String::new();
            let _ = self.stdin.read_line(&mut line);
            for w in line.split_whitespace() {
                self.buffer.push_back(String::from(w));
            }
        }
    }
    fn cin<T: FromStr>(&mut self) -> T {
        self.reserve();
        match self.buffer.pop_front().unwrap().parse::<T>() {
            Ok(a) => a,
            Err(_) => panic!("parse err")
        }
    }
}