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;

#define name ""
#define ini freopen(name".inp","r",stdin); freopen(name".out","w",stdout)
#define long long long
#define db double
#define pii pair <int, int>
#define pll pair <long, long>
#define all(c) c.begin(), c.end()
#define fastio ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0)
template <class T> inline void chmin(T &a, const T& val) { if (a > val) a = val; }
template <class T> inline void chmax(T &a, const T& val) { if (a < val) a = val; }
template <class T> long sqr(T a) { return 1ll * a * a; }
template <class T> void compress(T &a){
  sort(all(a)); a.resize(unique(all(a)) - a.begin());  
}

const db pi = acos(-1.00);
const db eps = 1e-6;
const int INF = 1e9;
const int MOD = 1e9+7;
const int dx[4] = {0, 0, -1, 1};
const int dy[4] = {-1, 1, 0, 0};

const int N = 3e3+1;
db p[N], dp[N][N];

int main()
{
  fastio;
  int n; cin >> n;
  for (int i = 1; i <= n; i++) cin >> p[i];
  dp[1][1] = p[1]; dp[1][0] = 1.00 - p[1];
  for (int i = 2; i <= n; i++){
    for (int j = 0; j <= i; j++){
      dp[i][j] += dp[i - 1][j] * (1.00 - p[i]);
      if (j > 0) dp[i][j] += dp[i - 1][j - 1] * p[i];
    }
  }
  db ans = 0;
  for (int i = 0; i <= n; i++) if (i > n - i) ans += dp[n][i];
  cout << fixed << setprecision(10) << ans;
  return 0;
}