Black Jack

AtCoder

IDabc342_f

Time2000ms

Memory256MB

Difficulty—

You will play a game against a dealer. The game uses a $D$\-sided die (dice) that shows an integer from $1$ to $D$ with equal probability, and two variables $x$ and $y$ initialized to $0$. The game proceeds as follows: * You may perform the following operation any number of times: roll the die and add the result to $x$. You can choose whether to continue rolling or not after each roll. * Then, the dealer will repeat the following operation as long as $y < L$: roll the die and add the result to $y$. * If $x > N$, you lose. Otherwise, you win if $y > N$ or $x > y$ and lose if neither is satisfied. Determine the probability of your winning when you act in a way that maximizes the probability of winning. ## Constraints * All inputs are integers. * $1 \leq L \leq N \leq 2 \times 10^5$ * $1 \leq D \leq N$ ## Input The input is given from Standard Input in the following format: $N$ $L$ $D$ [samples]

Samples

Input #1

3 2 2

Output #1

0.468750000000000

It can be proved that the optimal strategy is to continue rolling as long as $x$ is not greater than $2$.

Input #2

200000 200000 200000

Output #2

0.999986408692793

API Response (JSON)

{
  "problem": {
    "name": "Black Jack",
    "description": {
      "content": "You will play a game against a dealer. The game uses a $D$\\-sided die (dice) that shows an integer from $1$ to $D$ with equal probability, and two variables $x$ and $y$ initialized to $0$. The game 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": "abc342_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You will play a game against a dealer. The game uses a $D$\\-sided die (dice) that shows an integer from $1$ to $D$ with equal probability, and two variables $x$ and $y$ initialized to $0$. The game pr...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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