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<bits/stdc++.h>
using namespace std;
#define IOS ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define endl "\n"
#define int long long
float p[3000];
float dp[3000][1501];
int n;
float prob(int i,int heads)
{
    // cout<<"i: "<<i<<" heads: "<<heads<<" dp: "<<dp[i][heads]<<endl;
    if(dp[i][heads]>-1) return dp[i][heads];
    if(i==n-1) {if(heads>=1) dp[i][heads]= p[i]; else dp[i][heads]= 1;}
    else if(heads==n-i) dp[i][heads]= p[i]*prob(i+1,heads-1);
    else if(heads<n-i) {dp[i][heads]= p[i]*prob(i+1,heads-1)+(1-p[i])*prob(i+1,heads);}
    else
    {
        cout<<"heads: "<<heads<<" i: "<<n-i<<endl; 
        dp[i][heads]= 1;
    }
    return dp[i][heads];
}
int32_t main()
{
    IOS
    // #ifndef ONLINE_JUDGE
    // freopen("input.txt", "r", stdin);
    // freopen("output.txt", "w", stdout);
    // #endif
    cin>>n;
    for(int i=0;i<n;i++) cin>>p[i];
    for(int i=0;i<n;i++) for(int j=0;j<=(n%2?n/2+1:n/2);j++) dp[i][j]=-1;
    cout<<setprecision(8)<<prob(0,n%2?n/2+1:n/2);
    return 0;
}