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



Sample input:
3
0.30 0.60 0.80


Sample output:
0.612

Submission status: Runtime Error

Solution code:
import numpy as np
n=int(input())
s=list(map(float,input().split()))
a=[0 for _ in range(n+1)]
p = [0 for _in range(n+1)]
a[0]=1
for i in range(n):
    k=s[i]
    for j in range(len(a)):
        p[j] = a[j]*(1-k)
    for j in range(1,len(a)):
        p[j] += a[j-1]*k
    a = p
print(sum(a[n//2+1:]))

Submission status: Runtime Error

Solution code:
n = int(input())
p = list(map(float, input().split()))

dp = [[-1]*n for i in range(n)]

def solve(i, heads, tails):
    if (i==len(p)):
        if (heads>tails):
            return 1
        else:
            return 0

    if (dp[i][heads]!=-1):
        return dp[i][heads]

    dp[i][heads] = p[i]*solve(i+1, heads+1, tails)+(1-p[i])*solve(i+1, heads, tails+1)
    return dp[i][heads]

print(solve(0, 0, 0))