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 JavaScript (node.js v5.12) (judge verdict: Wrong Answer)

'use strict';
(function(input){
  input=input.trim().split(/\s+/).map(x=>x-0);
  const N=input[0], p=input[1];
  const q=1-p;
  //console.log(p,q);
  const dp=new Array(N+3).fill(0).map((x,i)=>new Array(i+1).fill(0));
  //console.log(dp);
  dp[2][1]=1;
  for(let i=3;i<=N+2;i++){
    for(let j=1;j<=i-1;j++){
      dp[i][j]=p*dp[i-1][j-1]+q*dp[i-1][j];
    }
  }
  //console.log(dp);
  var ans=0;
  for(let j=(N+3)/2;j<=N+1;j++){
    ans+=dp[N+2][j];
  }
  console.log(ans);
})(require('fs').readFileSync('/dev/stdin','utf8'));