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)]

#[macro_use]
mod _input {
    use std::fmt;
    use std::io::{self, Read};
    use std::str::{self, FromStr};

    macro_rules! input {
        ($($name:ident : $t:tt),*,) => {
            input!($($name: $t),*)
        };
        ($($name:ident : $t:tt),*) => {
            let mut _scanned = ::_input::Scanned::new(::std::io::stdin(), 1024);
            $(
                let $name = _read_value!(_scanned, $t);
            )*
        };
    }

    macro_rules! _read_value {
        ($scanned:expr, ($($t:tt),*)) => {
            ($(_read_value!($scanned, $t)),*)
        };
        ($scanned:expr, [$t:tt; $n:expr]) => {
            (0..$n).map(|_| _read_value!($scanned, $t)).collect::<Vec<_>>()
        };
        ($scanned:expr, _bytes) => {
            _read_value!($scanned, String).into_bytes()
        };
        ($scanned:expr, _1based) => {
            _read_value!($scanned, usize) - 1
        };
        ($scanned:expr, $t:ty) => {
            $scanned.next::<$t>()
        };
    }

    #[derive(Default)]
    pub struct Scanned {
        buf: Vec<u8>,
        pos: usize,
    }

    impl Scanned {
        pub fn new<R: Read>(mut input: R, estimated: usize) -> Self {
            let mut buf_ = Vec::with_capacity(estimated);
            let _ = io::copy(&mut input, &mut buf_).unwrap();
            if buf_.last() != Some(&b'\n') {
                buf_.push(b'\n');
            }
            debug_assert!(str::from_utf8(&buf_).is_ok());
            debug_assert!(buf_.iter().all(|&c| u32::from(c) <= 0x7F));
            Scanned { buf: buf_, pos: 0 }
        }

        #[inline]
        pub fn next<T: FromStr>(&mut self) -> T
        where
            T::Err: fmt::Debug,
        {
            let mut start = None;
            loop {
                match (self.buf[self.pos], start.is_some()) {
                    (b' ', true) | (b'\n', true) => break,
                    (_, true) | (b' ', false) | (b'\n', false) => self.pos += 1,
                    (_, false) => start = Some(self.pos),
                }
            }
            let target = &self.buf[start.unwrap()..self.pos];
            unsafe { str::from_utf8_unchecked(target) }.parse().unwrap()
        }
    }
}

use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::{self, BufWriter, Read, Stderr, StderrLock, Stdout, StdoutLock, Write};
use std::num::{ParseFloatError, ParseIntError};
use std::ops::{
    Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Div, DivAssign,
    Mul, MulAssign, Neg, Not, Rem, RemAssign, Sub, SubAssign,
};
use std::str::{self, FromStr};
use std::{char, cmp, f32, f64, fmt, i16, i32, i64, i8, mem, thread, u16, u32, u64, u8};

fn main() {
    input! {
        n: usize,
        ps: [f64; n],
    }
    let mut dp = vec![vec![0.0; n + 1]; n + 1];
    dp[0][0] = 1.0;
    for i in 0..n {
        dp[i + 1][0] = dp[i][0] * ps[i];
    }
    for j in 0..n {
        dp[0][j + 1] = dp[0][j] * (1.0 - ps[j]);
    }
    for i in 1..n {
        for j in 1..n {
            if i + j <= n {
                dp[i][j] = dp[i - 1][j] * ps[i + j - 1] + dp[i][j - 1] * (1.0 - ps[i + j - 1]);
            }
        }
    }
    let e = (0..n + 1)
        .filter(|&i| i > n - i)
        .map(|i| dp[i][n - i])
        .sum::<f64>();
    println!("{}", e);
}