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++ (GCC 9.2.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
//#include <boost/multiprecision/cpp_int.hpp>
//using namespace boost::multiprecision;
#define ll long long int
#define ar array
#define ld long double
#define bigint int64_t
#define vll vector<ll>
#define pll ar<ll,3>
#define vpll vector<pll>
#define pb push_back
#define sz(x) (int)(x).size()
#define ff first
#define MOD 1000000007
#define ss second
#define en cout<<"\n"
#define LCM(a,b) (a*(b/__gcd(a,b)))
#define all(v) v.begin(),v.end()
#define maxqueue    priority_queue<int>
#define minqueue    priority_queue<int,vector<int>,greater<int> >
#define lb(v,val)  (lower_bound(v.begin(),v.end(),val)-v.begin())
#define ub(v,val)  (upper_bound(v.begin(),v.end(),val)-v.begin())
#define INF (ll)(1e18)
#define setbits(x)      __builtin_popcountll(x)
#define zrobits(x)      __builtin_ctzll(x)
#define ini(a,b) memset(a,b,sizeof(a))
#define vvll vector<vector<ll>>
#define fast ios::sync_with_stdio(0);cin.tie(0);cout.tie(0);
#define MEM(a, b) memset(a, b, sizeof(a))
#define MAXN (int)(1 * 1e6 + 10)
#define prec(n) fixed<<setprecision(n)
#define deb1(x)                cerr<<#x<<": "<<x<<endl
#define deb2(x, y)             cerr<<#x<<": "<<x<<" | "<<#y<<": "<<y<<endl
#define deb3(x, y, z)          cerr<<#x<<":" <<x<<" | "<<#y<<": "<<y<<" | "<<#z<<": "<<z<<endl
#define deb4(a, b, c, d)       cerr<<#a<<": "<<a<<" | "<<#b<<": "<<b<<" | "<<#c<<": "<<c<<" | "<<#d<<": "<<d<<endl
#define deb5(a, b, c, d, e)    cerr<<#a<<": "<<a<<" | "<<#b<<": "<<b<<" | "<<#c<<": "<<c<<" | "<<#d<<": "<<d<<" | "<<#e<< ": "<<e<<endl
#define deb6(a, b, c, d, e, f) cerr<<#a<<": "<<a<<" | "<<#b<<": "<<b<<" | "<<#c<<": "<<c<<" | "<<#d<<": "<<d<<" | "<<#e<< ": "<<e<<" | "<<#f<<": "<<f<<endl
#define GET5(a, b, c, d, e, ...) e
#define D_EBC(...) GET5(__VA_ARGS__, deb4, deb3, deb2, deb1)
#define deb(...) D_EBC(__VA_ARGS__)(__VA_ARGS__)
#define each(x, a) for (auto& x: a)
#define F_OR(i, a, b, s) for (int i = (a); ((s) > 0 ? i < (b) : i > (b)); i += (s))
#define F_OR1(e) F_OR(i, 0, e, 1)
#define F_OR2(i, e) F_OR(i, 0, e, 1)
#define F_OR3(i, b, e) F_OR(i, b, e, 1)
#define F_OR4(i, b, e, s) F_OR(i, b, e, s)
#define GET5(a, b, c, d, e, ...) e
#define F_ORC(...) GET5(__VA_ARGS__, F_OR4, F_OR3, F_OR2, F_OR1)
#define forn(...) F_ORC(__VA_ARGS__)(__VA_ARGS__)
 
using namespace std;
 
// #include <ext/pb_ds/assoc_container.hpp>
// #include <ext/pb_ds/tree_policy.hpp>
// using namespace __gnu_pbds;
// template <typename T> using oset = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
// template <typename T> using mset = tree<T, null_type, less_equal<T>, rb_tree_tag, tree_order_statistics_node_update>;
// find_by_order(k) : iterator to value at kth index in sorted array. 
// order_of_key(k) : Index of the value k
 
template <typename T>
void read(T &x) {
    cin >> x;
}
template <typename T , typename T0>
void read(T &x, T0 &y) {
    cin >> x >> y;
}
 
template <typename T , typename T0 , typename T1>
void read(T &x, T0 &y, T1 &z) {
    cin >> x >> y >> z;
}
 
template <typename T , typename T0 , typename T1 , typename T2>
void read(T &x, T0 &y, T1 &z, T2 &w) {
    cin >> x >> y >> z >> w;
}
 
template <typename T , typename T0>
void read(pair< T , T0 > &p) {
    cin >> p.fst >> p.scd;
}
 
template <typename T>
void read(vector< T > &oneD, ll n) {
    for (ll i = 0; i < n; i++) {
        read(oneD[i]);
    }
 
}
 
template <typename T>
void read(T oneD[] , ll n) {
    for (ll i = 0; i < n; i++) {
        read(oneD[i]);
    }
}
 
template <typename T>
void write(T &x) {
 
    cout << x << " ";
}
 
template <typename T , typename T0>
void write(T &x, T0 &y) {
    cout << x << " " << y << "\n";
}
 
template <typename T , typename T0 , typename T1>
void write(T &x, T0 &y, T1 &z) {
    cout << x << " " << y << " " << z << "\n";
}
 
template <typename T , typename T0 , typename T1 , typename T2>
void write(T &x, T0 &y, T1 &z, T2 &w) {
    cout << x << " " << y << " " << z << " " << w << "\n";
}
 
template <typename T , typename T0>
void write(pair< T , T0 > &p) {
    write(p.fst);
    write(p.scd);
    cout << endl;
}
 
template <typename T>
void write(vector< T > &oneD) {
    for (ll i = 0; i < oneD.size(); i++) {
        write(oneD[i]);
    }
    cout << endl;
}
 
template <typename T>
void write(T oneD[] , int n) {
    for (ll i = 0; i < n; i++) {
        write(oneD[i]);
    }
    cout << endl;
}
 
template <typename T , typename T0>
void write(map< T , T0 > &mpp) {
    for (auto it : mpp) {
        write(it.fst);
        cout << ": ";
        write(it.scd);
        cout << "\n";
    }
    cout << endl;
}
template <typename T> inline bool umax(T &x, T y) {return (y > x) ? x = y, true : false;}
template <typename T> inline bool umin(T &x, T y) {return (y < x) ? x = y, true : false;}
 
ll FIRSTTRUE(function<bool(ll)> f, ll lb, ll rb) {
    while(lb<rb) {
        ll mb=(lb+rb)/2;
        f(mb)?rb=mb:lb=mb+1; 
    } 
    return lb;
}
ll LASTTRUE(function<bool(ll)> f, ll lb, ll rb) {
    while(lb<rb) {
        ll mb=(lb+rb+1)/2;
        f(mb)?lb=mb:rb=mb-1; 
    } 
    return lb;
}
 
inline bool isSquare(ll x)
{
    ll s = sqrt(x);
    return (s * s == x);
}
inline bool isPowerOfTwo(ll x)
{
    return ((1LL << (ll)log2(x)) == x);
}
 
ll extended_GCD(ll a , ll b , ll &x , ll &y) {
    if (a == 0) {
        x = 0;
        y = 1;
        return b;
    }
    ll x1 , y1;
    ll gcd = extended_GCD(b % a , a , x1 , y1);
    x = y1 - (b / a) * x1;
    y = x1;
    return gcd;
}
 
vector<ll> prime;
 
void sieve(ll N){
    bool isPrime[N+!1];
 
    for(ll i = 0; i <= N;++i) {
        isPrime[i] = true;
    }
    isPrime[0] = false;
    isPrime[1] = false;
    for(ll i = 2; i * i <= N; ++i) {
            if(isPrime[i] == true) {   
                for(ll j = i * i; j <= N ;j += i)
                    isPrime[j] = false;
        }
    }
 
    for(ll i =2;i<=N;i++) {
        if(isPrime[i] == true) {
            prime.push_back(i);
        }
    }
}
 
 
ll minPrime[MAXN];
void factorSieve(ll n){
    memset(minPrime,0,sizeof(minPrime));
    for (ll i = 2; i * i <= n; ++i) {
        if (minPrime[i] == 0) {
            for (ll j = i * i; j <= n; j += i) {
                if (minPrime[j] == 0) {
                    minPrime[j] = i;
                }
            }
        }
    }
    for (ll i = 2; i <= n; ++i) {
        if (minPrime[i] == 0) {
            minPrime[i] = i;
        }
    }
}
 
ll factorize(ll n) {
 
    map<ll,ll> m;
    m.clear();
    while (n != 1) {
        m[minPrime[n]]++;
        n /= minPrime[n];
    }
 
    for(auto i:m) {
        if(i.second>1) {
            return 0;
        }
    }
 
    return 1;
}
ll modinv(ll a , ll mod = MOD) {
    ll x , y, g;
    g = extended_GCD(a , mod , x , y);
    if (g != 1)
        return -1;
    else
    {
        if (x < 0) x += mod;
        return x;
    }
}
 
ll power(ll a, ll b, ll m = MOD) {
    a %= m;
    ll res = 1;
    while (b > 0) {
        if (b & 1)
            res = res * a % m;
        a = a * a % m;
        b >>= 1;
    }
    return res;
}
 
 
string to_string(char c) {
    return string(1, c);
}
 
struct comp {
bool operator()(pair<ll,ll> a, pair<ll,ll> b) {
    if(a.first == b.first) {
        return a.second > b.second;
    }
    return a.first < b.first;
}  
};
 
bool cmp (pair<ll,ll> a, pair<ll,ll> b) {
    if(a.first == b.first) {
        return a.second > b.second;
    }
    return a.first < b.first;
}  
 
// bool safe(ll i, ll j) { return i >= 0 && i < n && j >= 0 && j < m; }
const ll N=3e3;
ld dp[N+1][N+1],arr[N];
 
void solve(){
    ll n;
    read(n);
    forn(i,n)
    read(arr[i]);
    dp[0][0]=1;       //0 coins used and 0 head.
    forn(i,n){
        forn(j,0,i+1){
            dp[i+1][j]+=dp[i][j]*(1-arr[i]);
            dp[i+1][j+1]+=dp[i][j]*arr[i];
        }
    }
    ld ans=0;
    forn(i,(n+1)/2,n+1){
        ans+=dp[n][i];
    }
    cout<<prec(9)<<ans;
}
 
int main() {
    fast;
    int t=1;
    // read(t);
    forn(t) {
        // cout << "Case #" << i+1 << ": ";
        solve();
    }
}