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 C++14 (Clang 3.8.0) (judge verdict: Runtime Error)

#include<iostream>
#include<vector>
#include<string>
#include<algorithm>
#include<stack>
#include<unordered_set>
#include<cmath>
#include<tuple>
#include<numeric>
#include<iterator>
#include<set>
#include<unordered_map>
#include<map>
#include<queue>
#include<cstring>
#include<stdio.h>
#include<fstream>
#include<iomanip>

#define fastio ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL)
#define foru(i , a , b) for(ll i = a ; i < b ; i++)
#define ford(i , a , b) for(ll i = a ; i >= b ; i--)
#define ain(a , n) for(ll i = 0 ; i < n ; i++) cin >> a[i]
#define aout(a , n) for(ll i = 0 ; i < n ; i++) cout << a[i] << " "
#define M 1000000007
#define mp make_pair
#define mt make_tuple
#define cend cout << "\n"
#define all(x) x.begin() , x.end()
#define aout2(a , n , m) for(ll i = 0 ; i < n ; i++){for(ll j = 0 ; j < m ; j++){cout << a[i][j] << " ";} cout << endl;}
#define ain2(a , n , m) for(ll i = 0 ; i < n ; i++){for(ll j = 0 ; j < m ; j++){cin >> a[i][j];}}
#define ff first
#define ss second
#define mset(a , b) memset(a , b , sizeof(a))
#ifndef ONLINE_JUDGE
#define dbg(a) cerr << #a << ": "; _print(a); cerr << endl;
#else
#define dbg(a) // No more TLEs
#endif

using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef long double ldb;
typedef vector <int> vi;
typedef vector <ll> vll;
typedef vector <char> vch;
typedef vector <string> vs;
typedef set <ll> sll;
typedef multiset <ll> msll;
typedef pair <ll , ll> pll;
typedef vector <pair <ll , ll>> vpll;
void _print(ll t){cerr<<t;}
void _print(int t){cerr<<t;}
void _print(string t){cerr<<t;}
void _print(char t){cerr<<t;}
void _print(ldb t){cerr<<t;}
template <class T, class V> void _print(pair <T, V> p);
template <class T> void _print(vector <T> v);
template <class T> void _print(set <T> v);
template <class T, class V> void _print(map <T, V> v);
template <class T, class V> void _print(pair <T, V> p){cerr<< "{";_print(p.ff);cerr<<",";_print(p.ss);cerr<<"}";}
template <class T> void _print(vector <T> v){cerr<<"[ ";for(T i: v){_print(i);cerr<<" ";}cerr << "]";}
template <class T> void _print(set <T> v){cerr<<"[ ";for(T i: v){_print(i);cerr<<" ";}cerr << "]";}
template <class T, class V> void _print(map <T, V> v){cerr<<"[ ";for(auto i: v){_print(i);cerr<<" ";}cerr << "]";}

/*----------------------------- # --- MATH ALGORITHMS --- # -----------------------------*/

template <class T> T gcd(T a , T b){ while(a != 0){T temp = a; a = b % a; b = temp;}return b;}
template <class T> T egcd(T a , T b , T &x , T &y){T gcd , xt , yt;if(a == 0){gcd = b;x = 0 , y = 1;}else {gcd = egcd(b % a , a , xt , yt);x = yt - (b/a)*xt; y = xt;}return gcd;}
template <class T> T expo(T base , T exp , T mod){T res = 1;base = base % mod;while (exp > 0){if (exp & 1)res = (res*base) % mod;exp = exp>>1;base = (base*base) % mod;}return res;}
template <class T> T modinv(T a , T mod){T x , y; egcd<T>(a , mod , x , y);while(x < 0) x += mod; while(x >= mod) x -= mod; return x;}
template <class T> T modinvfermat(T a , T mod){return expo<T>(a , mod - 2 , mod);}
template <class T> bool rev(T a , T b){return a > b;}
template <class T> ll maxpower(T a , T b){ll ans = 0;while(a > 0 && a % b == 0){ans++;a /= b;}return ans;}
template <class T> T mceil(T a, T b){if(a % b == 0) return a/b; else return a/b + 1;}
template <class T> T lcm(T a, T b){return (a*b)/gcd<T>(a, b);}

/*----------------------------- # --- MAIN CODE --- # -----------------------------*/
int main() {
    fastio;
#ifndef ONLINE_JUDGE
    freopen("testcase.txt", "r", stdin);
    freopen("error.txt", "w", stderr);
    freopen("output.txt", "w", stdout);
#endif
    ll n; cin >> n;vector <vector <ldb>> dp(n + 1, vector <ldb> (n + 1, 0));
    vector <ldb> a(n); ain(a, n);
    dp[1][0] = 1 - a[0]; dp[1][1] = a[0];
    ldb prod = 1;
    foru(i, 0, n){
        prod *= (1 - a[i]);
        dp[i + 1][0] = prod;
    }
    foru(i, 2, n + 1){
        foru(j, 1, n + 1){
            dp[i][j] = dp[i - 1][j]*(1 - a[i - 1]) + dp[i - 1][j - 1]*a[i - 1];
        }
    }
    ldb ans = 0;
//    dbg(dp);
    foru(i, n/2 + 1, n + 1){
        ans += dp[n][i];
    }
    cout << setprecision(10);
    cout << ans;
}