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 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <climits>
#include <queue>
#include <vector>
#include <map>
#include <cstdlib>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <sstream>  // istringstream buffer(myString);
#include <stack>
#include <algorithm>  //swap function is here
#include <cstring>
#include <cassert>
#include <bits/stdc++.h>
using namespace std;


#define bit(x,i) (x&(1<<i))  //select the bit of position i of x
#define lowbit(x) ((x)&((x)^((x)-1))) //get the lowest bit of x
#define hBit(msb,n) asm("bsrl %1,%0" : "=r"(msb) : "r"(n)) //get the highest bit of x, maybe the fastest
#define max(a,b) (a<b?b:a)

#define IN(i,l,r) (l<i&&i<r) //the next for are for checking bound
#define LINR(i,l,r) (l<=i&&i<=r)
#define LIN(i,l,r) (l<=i&&i<r)
#define INR(i,l,r) (l<i&&i<=r)
#define F(i,L,R) for (int i = L; i < R; i++) //next four are for "for loops"
#define FE(i,L,R) for (int i = L; i <= R; i++)
#define FF(i,L,R) for (int i = L; i > R; i--)
#define FFE(i,L,R) for (int i = L; i >= R; i--)
#define getI(a) scanf("%d", &a) //next three are handy ways to get ints, it's also force you to use '&' sign
#define getII(a,b) scanf("%d%d", &a, &b)
#define getIII(a,b,c) scanf("%d%d%d", &a, &b, &c)
#define wez(n) int (n); scanf("%d",&(n)) //handy if the input is right after the definition of a variable
#define wez2(n,m) int (n),(m); scanf("%d %d",&(n),&(m))
#define wez3(n,m,k) int (n),(m),(k); scanf("%d %d %d",&(n),&(m),&(k))
#define TESTS wez(testow)while(testow--) //for multilple cases problems
#define whileZ int T; getI(T); while(T--) // the same as above
#define getS(x) scanf("%s", x) //get a char* string
#define clr(a,x) memset(a,x,sizeof(a)) //set elements of array to some value
#define char2Int(c) (c-'0')
#define lastEle(vec) vec[vec.size()-1]
#define SZ(x) ((int)((x).size()))
#define REMAX(a,b) (a)=max((a),(b)) // set a to the maximum of a and b
#define REMIN(a,b) (a)=min((a),(b));
#define FOREACH(i,t) for (typeof(t.begin()) i=t.begin(); i!=t.end(); i++) // traverse an STL data structure
#define ALL(c) (c).begin(),(c).end() //handy for function like "sort()"
#define PRESENT(c,x) ((c).find(x) != (c).end())
#define CPRESENT(c,x) (find(ALL(c),x) != (c).end())
#define ll long long //data types used often, but you don't want to type them time by time
#define ull unsigned long long
#define ui unsigned int
#define us unsigned short
#define IOS ios_base::sync_with_stdio(0); //to synchronize the input of cin and scanf
#define INF 1001001001
#define PI 3.1415926535897932384626
//for map, pair
#define mp make_pair
#define fi first
#define se second
// for debug
inline void pisz(int n) { printf("%d\n",n); }
#define DBG(vari) cerr<<#vari<<" = "<<(vari)<<endl;
#define printA(a,L,R) FE(i,L,R) cout << a[i] << (i==R?'\n':' ')
#define printV(a) printA(a,0,a.size()-1)
#define MAXN 10000
//for vectors
#define pb push_back
typedef int elem_t;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef pair<int,int> ii;
// directions
const int fx[4][2] = {{0,1}, {0,-1}, {1,0}, {-1,0}};
const int fxx[8][2] = {{0,1}, {0,-1}, {1,0}, {-1,0}, {1,1}, {1,-1}, {-1,1}, {-1,-1}};
template<typename T,typename TT> ostream& operator<<(ostream &s,pair<T,TT> t) {return s<<"("<<t.first<<","<<t.second<<")";}
template<typename T> ostream& operator<<(ostream &s,vector<T> t){F(i,0,SZ(t))s<<t[i]<<" ";return s; }
 
ll mod=1000000007;

ll gcd(ll a,ll b) 
{ 
    if (a == 0) 
        return b; 
    return gcd(b % a, a); 
} 

ll power(ll a,ll b,ll m)
{
    ll r=1;
    a=a%m;
    while(b)
    {
        if(b&1)r=r*a%m;
        a=a*a%m;
        b=b>>1;
    }
return r;
}


 float dp[3000][3000];

int main()
{


ios::sync_with_stdio(false);
 cin.tie(0);
 cout.tie(0);

 ll n;
 cin>>n;


 float p[n+1];

 for (int i = 1; i <= n; ++i)
 {
    cin>>p[i];
     /* code */
 }

 dp[0][0]=1;

 for (int i = 1; i <= n; ++i)
 {
    dp[i][0]=dp[i-1][0]*(1-p[i]);
     /* code */
 }
 for (int i = 1; i <= n; ++i)
  {

    for (int j = 1; j<= n; ++j)
    {
        dp[i][j]=dp[i-1][j]*(1-p[i])+dp[i-1][j-1]*p[i];

        /* code */
    }
      /* code */
  } 
float sum=0;

for (int i =n ; i >n/2; i--)
{
    sum=sum+dp[n][i];
    /* code */
}
cout<<sum<<"\n";


}