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)

use std::cmp::*;

#[allow(unused_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        eprintln!(concat!($(stringify!($a), " = {:?}, "),*), $($a),*);
    }
}

#[allow(unused_macros)]
macro_rules! max {
    ($x:expr) => {
      $x
    };
    ($x:expr, $($xs:tt)+) => {
      max($x,max!($($xs)+))
    };
  }

#[allow(unused_macros)]
macro_rules! min {
    ($x:expr) => {
      $x
    };
    ($x:expr, $($xs:tt)+) => {
      min($x,min!($($xs)+))
    };
  }

use std::ascii::AsciiExt;
use std::collections::HashMap;
use std::collections::VecDeque;
use std::io::Write;
use std::iter::Iterator;

#[derive(PartialEq, Debug, Clone)]
pub struct UnionFindNode {
    pub par: usize,
    pub rank: usize,
}

#[derive(PartialEq, Debug, Clone)]
pub struct UnionFind(pub Vec<UnionFindNode>);

impl UnionFind {
    //初期化
    pub fn new(size: usize) -> UnionFind {
        let mut vec = Vec::new();
        for i in 0..size {
            vec.push(UnionFindNode { par: i, rank: 0 });
        }

        UnionFind(vec)
    }

    //根を求める
    pub fn find(&mut self, x: usize) -> usize {
        if self.0[x].par == x {
            x
        } else {
            let par = self.0[x].par;
            let v = self.find(par);
            self.0[x].par = v;
            v
        }
    }

    //xとyの集合を併合
    pub fn unite(&mut self, x: usize, y: usize) {
        let x = self.find(x);
        let y = self.find(y);
        if x == y {
            return;
        }

        if self.0[x].rank < self.0[y].rank {
            self.0[x].par = y;
        } else {
            self.0[y].par = x;
            if self.0[x].rank == self.0[y].rank {
                self.0[x].rank += 1;
            }
        }
    }

    //xとyが同じ集合に属するか
    pub fn same(&mut self, x: usize, y: usize) -> bool {
        self.find(x) == self.find(y)
    }
}

pub trait LexicalPermutation {
    fn next_permutation(&mut self) -> bool;
    fn prev_permutation(&mut self) -> bool;
}

impl<T> LexicalPermutation for [T]
where
    T: Ord,
{
    fn next_permutation(&mut self) -> bool {
        if self.len() < 2 {
            return false;
        }
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] >= self[i] {
            i -= 1;
        }
        if i == 0 {
            return false;
        }
        let mut j = self.len() - 1;
        while j >= i && self[j] <= self[i - 1] {
            j -= 1;
        }
        self.swap(j, i - 1);
        self[i..].reverse();
        true
    }
    fn prev_permutation(&mut self) -> bool {
        if self.len() < 2 {
            return false;
        }
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] <= self[i] {
            i -= 1;
        }
        if i == 0 {
            return false;
        }
        self[i..].reverse();
        let mut j = self.len() - 1;
        while j >= i && self[j - 1] < self[i - 1] {
            j -= 1;
        }
        self.swap(i - 1, j);
        true
    }
}

//(小さい,同じ,大きい)
pub fn bound_count<T: Ord>(vec: &Vec<T>, val: &T) -> (usize, usize, usize) {
    let lb = lower_bound(vec, 0, vec.len(), val);
    let ub = upper_bound(vec, 0, vec.len(), val);
    (lb, ub - lb, vec.len() - ub)
}

//より小さい
pub fn less_count<T: Ord>(vec: &Vec<T>, val: &T) -> usize {
    lower_bound(vec, 0, vec.len(), val)
}

//より大きい
pub fn larger_count<T: Ord>(vec: &Vec<T>, val: &T) -> usize {
    vec.len() - upper_bound(vec, 0, vec.len(), val)
}

//同じ
pub fn eq_count<T: Ord>(vec: &Vec<T>, val: &T) -> usize {
    upper_bound(vec, 0, vec.len(), val) - lower_bound(vec, 0, vec.len(), val)
}

//vecは昇順ソート済み
//以上
pub fn lower_bound<T: Ord>(vec: &Vec<T>, mut first: usize, mut last: usize, val: &T) -> usize {
    let mut mid;
    while last - first > 1 {
        mid = (first + last) / 2;
        if &vec[mid] < val {
            first = mid;
        } else {
            last = mid;
        }
    }
    if &vec[first] < val {
        last
    } else {
        first
    }
}

//より大きい
pub fn upper_bound<T: Ord>(vec: &Vec<T>, mut first: usize, mut last: usize, val: &T) -> usize {
    let mut mid;
    while last - first > 1 {
        mid = (first + last) / 2;
        if &vec[mid] <= val {
            first = mid;
        } else {
            last = mid;
        }
    }
    if &vec[first] <= val {
        last
    } else {
        first
    }
}

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


    let mut dp = vec![vec![0.0; n + 2]; n + 2];

    dp[0][0] = 1.0;

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


    let mut ans = 0.0_f64;
    for j in n/2+1 .. n+1 {
        ans += dp[n][j];
    }

    println!("{}", ans);
}