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)

/* Simplicity and Goodness */

#include <bits/stdc++.h>
// #include <ext/pb_ds/assoc_container.hpp>
// #include <ext/pb_ds/tree_policy.hpp>

// using namespace __gnu_pbds;
using namespace std;
// typedef tree<int, null_type, less<int>, rb_tree_tag,
//             tree_order_statistics_node_update> indexed_set;

void my_dbg() { cout << endl; }
template<typename Arg, typename... Args> void my_dbg(Arg A, Args... B)
{ cout << ' ' << A; my_dbg(B...); }
#define dbg(...)  cout << "(" << #__VA_ARGS__ << "):", my_dbg(__VA_ARGS__)

#define scn(n) scanf("%d", &n)
#define lscn(n) scanf("%lld", &n)
#define pri(n) printf("%d ", (int)(n))
#define prin(n) printf("%d\n", (int)(n))
#define lpri(n) printf("%lld ", n)
#define lprin(n) printf("%lld\n", n)
#define rep(i,a,b) for(int i=(int)(a); i<(int)(b); i++)
#define pb push_back
#define mp make_pair
#define F first
#define S second

using ll = long long;
using vi = vector<int>;
using vl = vector<ll>;

const int inf = INT_MAX;
const int ninf = INT_MIN;
const int mod = 1e9+7;
const int N = 2e5+2;


void solve()
{
    int n;
    scn(n);

    vector<double> dp(n+1);
    // dp[i] : prob. of i heads so far

    dp[0] = 1;

    for(int coins = 0; coins < n; coins++) {
        double p_head;
        scanf("%lf", &p_head);

        for(int i = coins + 1; i >= 0; i--) {
            dp[i] = (i == 0? 0: dp[i-1] * p_head) + dp[i] * (1 - p_head);
        }   
    } 

    double ans = 0;

    for(int i=(n+1)/2; i<=n; i++) {
        ans += dp[i];
    }

    printf("%.12lf", ans);
}

int main()
{
    int t = 1;
    // scn(t);
    
    while(t --) {
        solve();
    }
    return 0;
}