Coins

AtCoder

IDdp_i

Time2000ms

Memory256MB

Difficulty—

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: $N$ $p_1$ $p_2$ $\ldots$ $p_N$ [samples]

Samples

Input #1

3
0.30 0.60 0.80

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 $0.144 + 0.336 + 0.096 + 0.036 = 0.612$.

Input #2

1
0.50

Output #2

0.5

Outputs such as `0.500`, `0.500000001` and `0.499999999` are also considered correct.

Input #3

5
0.42 0.01 0.42 0.99 0.42

Output #3

0.3821815872

API Response (JSON)

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  "problem": {
    "name": "Coins",
    "description": {
      "content": "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 pr",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "dp_i"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Let $N$ be a positive odd number.\nThere 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 pr...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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