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>
using namespace std;

#ifdef LOCAL
#define trace(...) __f(#__VA_ARGS__, __VA_ARGS__)
#else
#define trace(...) 42
#endif
template <typename Arg1>
void __f(const char* name, Arg1&& arg1){
    cerr << name << ": " << arg1 << endl;
}
template <typename Arg1, typename... Args>
void __f(const char* names, Arg1&& arg1, Args&&... args){
    const char* comma = strchr(names + 1, ',');
    cerr.write(names, comma - names) << ": " << arg1 << "\t|";
    __f(comma + 1, args...);
}

#define print(x) for(auto it : x) cout<< it << ' '; cout << '\n';
typedef long long ll;
typedef unsigned long long ull;
#define mem1(a)           memset(a,-1,sizeof(a))
#define mem0(a)           memset(a,0,sizeof(a))

const int MOD = 1e9 + 7;
ll mod(ll a) { return (ll) (a % MOD + MOD) % MOD; }
ll mul(ll a, ll b) { return (ll) mod(mod(a) * mod(b));}

// dp[index][heads] => probability of number of heads till this index
double dp[3000][3000];
double p[3000];

int main() {
    ios::sync_with_stdio(false); cin.tie(0);
    int n; cin >> n;
    for (int i = 0; i < n; i++)
        cin >> p[i];
    dp[0][0] = 1 - p[0];
    dp[0][1] = p[0];
    
    for (int i = 1; i < n; i++) {
        for (int j = 0; j <= n; j++) {
            // tails
            dp[i][j] = dp[i - 1][j] * (1 - p[i]);
            // heads
            if (j > 0)
                dp[i][j] += dp[i - 1][j - 1] * p[i];
        }
    }
    
//    for (int i = 0; i < n; i++) {
//        for (int j = 0; j < n; j++) {
//            cout << dp[i][j] << ' ';
//        }
//        cout << '\n';
//    }
    
    double ans = 0;
    for (int i = n/2 + 1; i <= n; i++) {
        ans += dp[n - 1][i];
    }
    cout << setprecision(9) << fixed << ans << '\n';
    return 0;
}