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: Accepted)

'use strict'

function main(input){
  const lines = input.split('\n');
  const N = parseInt(lines.shift());
  const P = lines.shift().split(' ').map(x => parseFloat(x));

//console.log([N, P]);

  const dp=[];
  for(var i =0;i<N;i++){
    dp.push(Array(N+2).fill(0.0))
  }

  dp[0][0]=P[0];
  dp[0][1]=1-P[0];

  for(var i =1;i<N;i++){
    for(var j=0;j<=i+1;j++){
      let pre_left = dp[i-1][j-1];
      if(pre_left==undefined){pre_left=0.0;}

      let pre_right = dp[i-1][j];
      if(pre_right==undefined){pre_right=0.0;}

      dp[i][j] = pre_left * (1.0 - P[i]) + pre_right * P[i]
//console.log([pre_left, pre_right]);

    }
//console.log('-')
  }


//console.log(dp);

  let result = 0;

  for(var i=0;i<N/2;i++){
    result += dp[N-1][i]
  }

  console.log(result);
}

main(require('fs').readFileSync('/dev/stdin', 'utf8'));