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: Time Limit Exceeded)

pub trait Readable {
    type Output;
    fn words_count() -> usize;
    fn read_words(words: &[&str]) -> Result<Self::Output, String>;
}
#[macro_export]
macro_rules! readable {
    ( $ t : ty , $ words_count : expr , |$ words : ident | $ read_words : expr ) => {
        impl Readable for $t {
            type Output = $t;
            fn words_count() -> usize {
                $words_count
            }
            fn read_words($words: &[&str]) -> Result<$t, String> {
                Ok($read_words)
            }
        }
    };
}
readable!((), 1, |_ss| ());
readable!(String, 1, |ss| ss[0].to_string());
impl Readable for char {
    type Output = char;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<char, String> {
        let chars: Vec<char> = words[0].chars().collect();
        if chars.len() == 1 {
            Ok(chars[0])
        } else {
            Err(format!("cannot parse `{}` as a char", words[0]))
        }
    }
}
pub struct Chars();
impl Readable for Chars {
    type Output = Vec<char>;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<Vec<char>, String> {
        Ok(words[0].chars().collect())
    }
}
macro_rules ! impl_readable_for_ints { ( $ ( $ t : ty ) * ) => { $ ( impl Readable for $ t { type Output = Self ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result <$ t , String > { use std :: str :: FromStr ; <$ t >:: from_str ( words [ 0 ] ) . map_err ( | _ | { format ! ( "cannot parse `{}` as {}" , words [ 0 ] , stringify ! ( $ t ) ) } ) } } ) * } ; }
impl_readable_for_ints ! ( i8 u8 i16 u16 i32 u32 i64 u64 isize usize f32 f64 ) ;
macro_rules ! define_one_origin_int_types { ( $ new_t : ident $ int_t : ty ) => { # [ doc = " Converts 1-origin integer into 0-origin when read from stdin." ] # [ doc = "" ] # [ doc = " # Example" ] # [ doc = "" ] # [ doc = " ```no_run" ] # [ doc = " # #[macro_use] extern crate atcoder_snippets;" ] # [ doc = " # use atcoder_snippets::read::*;" ] # [ doc = " // Stdin: \"1\"" ] # [ doc = " read!(a = usize_);" ] # [ doc = " assert_eq!(a, 0);" ] # [ doc = " ```" ] # [ allow ( non_camel_case_types ) ] pub struct $ new_t ; impl Readable for $ new_t { type Output = $ int_t ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { <$ int_t >:: read_words ( words ) . map ( | n | n - 1 ) } } } ; ( $ new_t : ident $ int_t : ty ; $ ( $ inner_new_t : ident $ inner_int_t : ty ) ;* ) => { define_one_origin_int_types ! ( $ new_t $ int_t ) ; define_one_origin_int_types ! ( $ ( $ inner_new_t $ inner_int_t ) ;* ) ; } ; }
define_one_origin_int_types ! ( u8_ u8 ; u16_ u16 ; u32_ u32 ; u64_ u64 ; usize_ usize ) ;
macro_rules ! impl_readable_for_tuples { ( $ t : ident $ var : ident ) => ( ) ; ( $ t : ident $ var : ident ; $ ( $ inner_t : ident $ inner_var : ident ) ;* ) => { impl_readable_for_tuples ! ( $ ( $ inner_t $ inner_var ) ;* ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,*> Readable for ( $ t , $ ( $ inner_t ) ,* ) { type Output = ( <$ t >:: Output , $ ( <$ inner_t >:: Output ) ,* ) ; fn words_count ( ) -> usize { let mut n = <$ t >:: words_count ( ) ; $ ( n += <$ inner_t >:: words_count ( ) ; ) * n } # [ allow ( unused_assignments ) ] fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { let mut start = 0 ; let $ var = <$ t >:: read_words ( & words [ start .. start +<$ t >:: words_count ( ) ] ) ?; start += <$ t >:: words_count ( ) ; $ ( let $ inner_var = <$ inner_t >:: read_words ( & words [ start .. start +<$ inner_t >:: words_count ( ) ] ) ?; start += <$ inner_t >:: words_count ( ) ; ) * Ok ( ( $ var , $ ( $ inner_var ) ,* ) ) } } } ; }
impl_readable_for_tuples ! ( T8 x8 ; T7 x7 ; T6 x6 ; T5 x5 ; T4 x4 ; T3 x3 ; T2 x2 ; T1 x1 ) ;
pub trait ReadableFromLine {
    type Output;
    fn read_line(line: &str) -> Result<Self::Output, String>;
}
fn split_into_words(line: &str) -> Vec<&str> {
    #[allow(deprecated)]
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: Readable> ReadableFromLine for T {
    type Output = T::Output;
    fn read_line(line: &str) -> Result<T::Output, String> {
        let words = split_into_words(line);
        if words.len() != T::words_count() {
            return Err(format!(
                "line `{}` has {} words, expected {}",
                line,
                words.len(),
                T::words_count()
            ));
        }
        T::read_words(&words)
    }
}
macro_rules ! impl_readable_from_line_for_tuples_with_from_iterator { ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident ) => { impl <$ u : Readable > ReadableFromLine for $ seq_in where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = $ seq_out ; fn read_line ( line : & str ) -> Result <$ seq_out , String > { let n = $ u :: words_count ( ) ; let words = split_into_words ( line ) ; if words . len ( ) % n != 0 { return Err ( format ! ( "line `{}` has {} words, expected multiple of {}" , line , words . len ( ) , n ) ) ; } let mut result = Vec :: new ( ) ; for chunk in words . chunks ( n ) { match $ u :: read_words ( chunk ) { Ok ( v ) => result . push ( v ) , Err ( msg ) => { let flagment_msg = if n == 1 { format ! ( "word {}" , result . len ( ) ) } else { let l = result . len ( ) ; format ! ( "words {}-{}" , n * l + 1 , ( n + 1 ) * l ) } ; return Err ( format ! ( "{} of line `{}`: {}" , flagment_msg , line , msg ) ) ; } } } Ok ( result . into_iter ( ) . collect ( ) ) } } impl < T : Readable , $ u : Readable > ReadableFromLine for ( T , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( T :: Output , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let n = T :: words_count ( ) ; # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; Ok ( ( T :: read_words ( words ) ?, <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident , $ ( $ inner_t : ident $ inner_var : ident ) ,+ ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ seq_in , $ seq_out ; $ ( $ inner_t $ inner_var ) ,+ ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,+ , $ u : Readable > ReadableFromLine for ( $ t , $ ( $ inner_t ) ,+ , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( $ t :: Output , $ ( $ inner_t :: Output ) ,+ , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let mut n = $ t :: words_count ( ) ; $ ( n += $ inner_t :: words_count ( ) ; ) + # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; let ( $ var , $ ( $ inner_var ) ,* ) = < ( $ t , $ ( $ inner_t ) ,+ ) >:: read_words ( words ) ?; Ok ( ( $ var , $ ( $ inner_var ) ,* , <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; }
#[macro_export]
macro_rules ! readable_collection { ( $ u : ident => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 t4 , T3 t3 , T2 t2 , T1 t1 ) ; } ; ( $ u : ident : $ ( $ bound : path ) ,* => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 t4 , T3 t3 , T2 t2 , T1 t1 ) ; } }
readable_collection ! ( U => Vec < U >, Vec < U :: Output > ) ;
readable_collection ! ( U : Eq , std :: hash :: Hash => std :: collections :: HashSet < U >, std :: collections :: HashSet < U :: Output > ) ;
pub fn read<T: ReadableFromLine>() -> T::Output {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();
    T::read_line(&line).unwrap()
}
#[macro_export]
macro_rules ! read { ( ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; } ; ( $ pat : pat = $ t : ty ) => { let $ pat = read ::<$ t > ( ) ; } ; ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { read ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ) ; } ; }
#[macro_export]
macro_rules ! readls { ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { $ ( read ! ( $ pat = $ t ) ; ) * } ; }
pub fn readx<T: ReadableFromLine>() -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result = stdin
        .lock()
        .lines()
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    result
}
#[macro_export]
macro_rules ! readx_loop { ( |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; for line in stdin . lock ( ) . lines ( ) { let line = line . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } ; ( |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readx_loop ! ( | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub fn readn<T: ReadableFromLine>(n: usize) -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result: Vec<T::Output> = stdin
        .lock()
        .lines()
        .take(n)
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    if result.len() < n {
        panic!(
            "expected reading {} lines, but only {} lines are read",
            n,
            result.len()
        );
    }
    result
}
#[macro_export]
macro_rules ! readn_loop { ( $ n : expr , |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; { let mut lock = stdin . lock ( ) ; for _ in 0 ..$ n { let mut line = String :: new ( ) ; lock . read_line ( & mut line ) . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } } ; ( $ n : expr , |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readn_loop ! ( $ n , | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub trait Words {
    fn read<T: Readable>(&self) -> T::Output;
}
impl<'a> Words for [&'a str] {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(self).unwrap()
    }
}
impl<'a> Words for &'a str {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(&[self]).unwrap()
    }
}

pub trait WithZero:
Sized + Eq + for<'a> std::ops::Add<&'a Self, Output = Self> + for<'a> std::ops::AddAssign<&'a Self>
{
    fn zero() -> Self;
}
pub trait WithOne:
Sized + Eq + for<'a> std::ops::Mul<&'a Self, Output = Self> + for<'a> std::ops::MulAssign<&'a Self>
{
    fn one() -> Self;
}
pub trait Integer:
WithZero
    + WithOne
    + Ord
    + for<'a> std::ops::Sub<&'a Self, Output = Self>
    + for<'a> std::ops::SubAssign<&'a Self>
    + for<'a> std::ops::Div<&'a Self, Output = Self>
    + for<'a> std::ops::DivAssign<&'a Self>
{
}
macro_rules! forward_ref_unop {
    ( impl $ op : ident , $ method : ident for $ t : ty ) => {
        impl std::ops::$op for &$t {
            type Output = <$t as std::ops::$op>::Output;
            fn $method(self) -> <$t as std::ops::$op>::Output {
                std::ops::$op::$method(*self)
            }
        }
    };
}
macro_rules! forward_ref_binop {
    ( impl $ op : ident , $ method : ident for $ t : ty , $ u : ty ) => {
        impl<'a> std::ops::$op<$u> for &'a $t {
            type Output = <$t as std::ops::$op<$u>>::Output;
            fn $method(self, other: $u) -> <$t as std::ops::$op<$u>>::Output {
                std::ops::$op::$method(*self, other)
            }
        }
        impl std::ops::$op<&$u> for $t {
            type Output = <$t as std::ops::$op<$u>>::Output;
            fn $method(self, other: &$u) -> <$t as std::ops::$op<$u>>::Output {
                std::ops::$op::$method(self, *other)
            }
        }
        impl std::ops::$op<&$u> for &$t {
            type Output = <$t as std::ops::$op<$u>>::Output;
            fn $method(self, other: &$u) -> <$t as std::ops::$op<$u>>::Output {
                std::ops::$op::$method(*self, *other)
            }
        }
    };
}
macro_rules! forward_ref_op_assign {
    ( impl $ op : ident , $ method : ident for $ t : ty , $ u : ty ) => {
        impl std::ops::$op<&$u> for $t {
            fn $method(&mut self, other: &$u) {
                std::ops::$op::$method(self, *other);
            }
        }
    };
}
pub trait PrimitiveInteger:
Sized
    + Ord
    + std::ops::Add<Output = Self>
    + std::ops::Sub<Output = Self>
    + std::ops::Div<Output = Self>
{
    fn one() -> Self;
    fn abs_diff(self, rhs: Self) -> Self;
    fn rem_euclid(self, rhs: Self) -> Self;
}
macro_rules ! impl_primitive_integer { ( $ ( $ t : ty ) * ) => { $ ( impl PrimitiveInteger for $ t { fn one ( ) -> $ t { 1 } fn abs_diff ( self , rhs : $ t ) -> $ t { if self < rhs { rhs - self } else { self - rhs } } # [ allow ( unused_comparisons ) ] fn rem_euclid ( self , rhs : $ t ) -> $ t { let r = self % rhs ; if r < 0 { if rhs < 0 { r - rhs } else { r + rhs } } else { r } } } ) * } }
impl_primitive_integer ! ( u8 u16 u32 u64 usize i8 i16 i32 i64 isize ) ;
pub trait PrimitiveUnsigned: PrimitiveInteger {
    fn ceil_div(self, rhs: Self) -> Self;
    fn round_div(self, rhs: Self) -> Self;
    fn log2(self) -> Option<Self>;
    fn ceil_log2(self) -> Option<Self>;
    fn sqrt(self) -> Self;
}
macro_rules ! impl_primitive_unsigned { ( $ ( $ t : ty ) * ) => { $ ( impl PrimitiveUnsigned for $ t { fn ceil_div ( self , rhs : $ t ) -> $ t { ( self + rhs - 1 ) / rhs } fn round_div ( self , rhs : $ t ) -> $ t { ( self + rhs / 2 ) / rhs } fn log2 ( mut self ) -> Option <$ t > { if self == 0 { None } else { let mut ans = 0 ; while self > 1 { ans += 1 ; self /= 2 ; } Some ( ans ) } } fn ceil_log2 ( self ) -> Option <$ t > { self . log2 ( ) . map ( | x | { ( self + ( ( 1 << x ) - 1 ) ) . log2 ( ) . unwrap ( ) } ) } fn sqrt ( self ) -> $ t { ( self as f64 ) . sqrt ( ) as $ t } } ) * } }
impl_primitive_unsigned ! ( u8 u16 u32 u64 usize ) ;

fn solve(ps: &[f64], n: usize, memo: &mut [Option<f64>]) -> f64 {
    if n == 0 { 1.0 } else { match ps.split_first() {
        None => 0.0,
        Some((&hd, tl)) => hd * solve(tl, n - 1, memo) + (1.0 - hd) * solve(tl, n, memo)
    }}
}

fn main() {
    read!(n = usize);
    read!(ps = Vec<f64>);
    let over_half = (n+1).ceil_div(2);
    println!("{}", solve(&ps, over_half, &mut vec![None; over_half + 1]));
}