Merge pull request #18 from shenxiangzhuang/add/fifty/chuck_a_luck

Added:fifty:6_chuck_a_lick

docs/fifty/6_chuck_a_luck.md

@@ -0,0 +1,108 @@
+# Chuck-a-Luck
+
+## Problem
+
+???+ question "Question"
+
+    === "English"
+
+        Chuck-a-Luck is a gambling game often played at carnivals and gambling houses.
+        A player may bet on anyone of the numbers 1,2,3,4,5,6.
+        Three dice are rolled. If the player's number appears on one, two, or three of the dice,
+        he receives respectively one, two, or three times his original stake plus his own money back;
+        otherwise he loses his stake. What is the player's expected loss per unit stake?
+        (Actually the player may distribute stakes on several numbers,
+        but each such stake can be regarded as a separate bet.)
+
+    === "中文"
+
+        Chuck-a-Luck是一种常在嘉年华和赌场中玩的赌博游戏。
+        玩家可以押注1、2、3、4、5或6中的任意一个数字。三个骰子被投掷，如果玩家押注的数字出现在一个、两个或三个骰子中，
+        他将分别获得其原始赌注的一倍、两倍或三倍以及自己的本金；否则他将输掉他的赌注。
+        玩家每投注一单位，他的预期损失是多少？（实际上，玩家可以在几个数字上分配赌注，但每个赌注可以视为一次单独的押注。）
+
+
+## Tip
+
+??? tip "Tip"
+
+    === "English"
+
+        Since we are looking for expectations, we need to first define the random variable,
+        then find out the probability of its distribution, and finally find the expectations.
+
+    === "中文"
+
+        既然是求期望，那么我们需要先定义随机变量，然后找出其分布的概率，最后求期望即可。
+
+
+## Solutions
+
+### Solution1: Analysis
+
+??? success "Solution1: Analysis"
+
+    === "English"
+
+        Let $R$ be the player's reward and $S$ be the original stake. Then the player's expected reward is $E(R)$, and the expected loss is $E(L) = -E(R)$.
+
+        To find $E(R)$, we first list the distribution of $R$ as follows:
+
+        |   | $R$   | Probability | Calculation Result |
+        |---|-------|-------------|--------------------|
+        | 1 | $-S$  | $\left[\frac{5}{6}\right]^3$ | $\frac{125}{216}$ |
+        | 2 | $S$   | $\frac{1}{6}\cdot\frac{5}{6}\cdot\frac{5}{6} + \frac{5}{6}\cdot\frac{1}{6}\cdot\frac{5}{6} + \frac{5}{6}\cdot\frac{5}{6}\cdot\frac{1}{6}$ | $\frac{75}{216}$ |
+        | 3 | $2S$  | $\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{5}{6} + \frac{1}{6}\cdot\frac{5}{6}\cdot\frac{1}{6} + \frac{5}{6}\cdot\frac{1}{6}\cdot\frac{1}{6}$ | $\frac{15}{216}$ |
+        | 4 | $3S$  | $\left[\frac{1}{6}\right]^3$ | $\frac{1}{216}$ |
+
+        From this, we can compute:
+
+        $$
+        E(R) = -S\cdot\frac{125}{216} + S\cdot\frac{75}{216} + 2S\cdot\frac{15}{216} + 3S\cdot\frac{1}{216} = -\frac{17}{216}S
+        $$
+
+        $$
+        E(L) = -E(R) = \frac{17}{216}S
+        $$
+
+    === "中文"
+
+        记玩家的奖励为$R$，原始的赌注为$S$，那么玩家的预期奖励为$E(R)$,
+        预期的损失为$E(L) = -E(R)$.
+        因为要求$E(R)$，所以我们先列出$R$的分布，如下：
+
+        |   | $R$   | Probability | Calculation Result |
+        |---|-------|-------------|--------------------|
+        | 1 | $-S$  | $\left[\frac{5}{6}\right]^3$ | $\frac{125}{216}$ |
+        | 2 | $S$   | $\frac{1}{6}\cdot\frac{5}{6}\cdot\frac{5}{6} + \frac{5}{6}\cdot\frac{1}{6}\cdot\frac{5}{6} + \frac{5}{6}\cdot\frac{5}{6}\cdot\frac{1}{6}$ | $\frac{75}{216}$ |
+        | 3 | $2S$  | $\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{5}{6} + \frac{1}{6}\cdot\frac{5}{6}\cdot\frac{1}{6} + \frac{5}{6}\cdot\frac{1}{6}\cdot\frac{1}{6}$ | $\frac{15}{216}$ |
+        | 4 | $3S$  | $\left[\frac{1}{6}\right]^3$ | $\frac{1}{216}$ |
+
+        由此计算出
+
+        $$
+            E(R) = -S\cdot\frac{125}{216} + S\cdot\frac{75}{216} + 2S\cdot\frac{15}{216} + 3S\cdot\frac{1}{216} = -\frac{17}{216}S
+        $$
+
+        $$
+            E(L) = -E(R) = \frac{17}{216}S
+        $$
+
+### Solution2: Simulation
+
+??? success "Solution2: Simulation"
+
+    === "English"
+
+        We can also simulate the game to find the expected loss:
+        ```python exec="true" source="material-block" session="fifty-6"
+        --8<-- "docs/fifty/snippet/6_chuck_a_luck.py:solution2"
+        ```
+
+    === "中文"
+
+        可以直接模拟整个赌博过程来获取近似的结果:
+
+        ```python exec="true" source="material-block" session="fifty-6"
+        --8<-- "docs/fifty/snippet/6_chuck_a_luck.py:solution2"
+        ```

docs/fifty/snippet/6_chuck_a_luck.py

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+# --8<-- [start:solution2]
+import random
+
+from typing import Dict, List, Tuple
+
+
+def bet() -> Tuple[float, int]:
+    """
+    A bet action
+    :return: the bet amount(stake) and the guess number
+    """
+    stake = 1.0
+    guess = random.randint(1, 6)
+    return stake, guess
+
+
+def toss_three() -> List[int]:
+    """
+    A toss action
+    :return: the three dice number
+    """
+    dices = [random.randint(1, 6) for _ in range(3)]
+    return dices
+
+
+def calc_loss(stake: float, guess: int, dices: List[int]) -> float:
+    """
+    Calculate the loss amount
+    :param stake: the stake amount
+    :param guess: the guess number
+    :param dices: the dice numbers
+    :return: the loss amount
+    """
+    guess_count = dices.count(guess)
+    if guess_count == 0:
+        return stake
+    else:
+        return -stake * guess_count
+
+
+def play() -> float:
+    """
+    Play a game
+    :return: the final amount
+    """
+    stake, guess = bet()
+    dices = toss_three()
+    loss = calc_loss(stake, guess, dices)
+    return loss
+
+
+def simulation(run_nums: int = 1000000, verbose: bool = True) -> float:
+    amounts: Dict[str, List] = {"amounts": []}
+    final_amount = 0.0
+    for _ in range(run_nums):
+        final_amount += play()
+        amounts["amounts"].append(final_amount)
+
+    expected_amount = final_amount / run_nums
+
+    if verbose:
+        print(f"Simulation expect loss amount: {expected_amount}\n")
+        print(f"Theory expect loss amount: 17/216 = {17 / 216}\n")
+    return expected_amount
+
+
+simulation()
+
+# --8<-- [end:solution2]

mkdocs.yml

@@ -16,6 +16,7 @@ nav:
     - "3. Flippant Jury": fifty/3_flippant_jury.md
     - "4. Trials until first success": fifty/4_first_success.md
     - "5. Coin in Square": fifty/5_coin_in_square.md
+    - "6. Chuck a Luck": fifty/6_chuck_a_luck.md
 
   - Changelog: CHANGELOG.md
 not_in_nav: |