飛鏢 Monte Carlo 模擬

本文根據 Frank R. Giordano 等人所著 A First Course in Mathematical Modeling 的一個專案，使用 Monte Carlo 方法模擬飛鏢遊戲。

飛鏢模擬

靶面區域及得分如下：

由原點——靶心中心——起計，各環半徑為：

靶面半徑為 1 ft，即 12 in.。我們需要先假設飛鏢落點的機率分佈，再寫出演算法。運行 1,000 次模擬，估計每局投五支飛鏢的平均得分，並找出期望價值——該環分數乘以命中機率——最高的環。

解法

先匯入所需函式庫並設定繪圖：

import numpy as np
import matplotlib.pyplot as plt

import matplotlib
matplotlib.rcParams.update({'font.family': 'serif', 'font.size': 16})

Monte Carlo 模擬仍應設定隨機種子，令結果可以重現：

np.random.seed(23)

定義靶面尺寸與各區分數：

BULLSEYE_RADIUS = 1.0
YELLOW_RADIUS = 2.5
BLUE_RADIUS = 5.0
RED_RADIUS = 8.0
WHITE_RADIUS = 12.0

POINTS = {
    "Bull's eye": 50,
    "Yellow ring": 25,
    "Blue ring": 15,
    "Red ring": 10,
    "White ring": 5,
    "Miss": 0
}

throw_dart 假設水平與垂直誤差互相獨立，並都服從以靶心為平均、標準差為 std_dev 的常態分佈。由此得到的徑向距離服從 Rayleigh 分佈：

def throw_dart(std_dev):
    """Simulate throwing a dart"""
    x = np.random.normal(0, std_dev)
    y = np.random.normal(0, std_dev)
    return x, y, np.sqrt(x**2 + y**2)

按距離決定每一投的得分：

def score_throw(distance):
    """Score a throw based on its distance from the center"""
    if distance <= BULLSEYE_RADIUS:
        return POINTS["Bull's eye"]
    elif distance <= YELLOW_RADIUS:
        return POINTS["Yellow ring"]
    elif distance <= BLUE_RADIUS:
        return POINTS["Blue ring"]
    elif distance <= RED_RADIUS:
        return POINTS["Red ring"]
    elif distance <= WHITE_RADIUS:
        return POINTS["White ring"]
    else:
        return POINTS["Miss"]

接着模擬多局遊戲，同時保存總分、各環命中次數與所有落點：

def run_simulation(num_simulations, num_darts, std_dev):
    """Run the Monte Carlo simulation"""
    total_scores = []
    ring_hits = {ring: 0 for ring in POINTS.keys()}
    all_throws = []

    for _ in range(num_simulations):
        game_score = 0
        for _ in range(num_darts):
            x, y, distance = throw_dart(std_dev)
            score = score_throw(distance)
            game_score += score
            all_throws.append((x, y))

            for ring, radius in zip(
                ["Bull's eye", "Yellow ring", "Blue ring", "Red ring", "White ring"],
                [BULLSEYE_RADIUS, YELLOW_RADIUS, BLUE_RADIUS, RED_RADIUS, WHITE_RADIUS]
            ):
                if distance <= radius:
                    ring_hits[ring] += 1
                    break
            else:
                ring_hits["Miss"] += 1

        total_scores.append(game_score)

    return total_scores, ring_hits, all_throws

繪圖函數同時顯示靶面落點與每局總分直方圖：

def plot_dartboard(all_throws):
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(20, 10))

    radii = [BULLSEYE_RADIUS, YELLOW_RADIUS, BLUE_RADIUS, RED_RADIUS, WHITE_RADIUS]
    colors = ['red', 'yellow', 'blue', 'red', 'white']

    for radius, color in zip(radii, colors):
        circle = plt.Circle((0, 0), radius, fill=False, color=color)
        ax1.add_artist(circle)

    x, y = zip(*all_throws)
    ax1.set_facecolor('black')
    ax1.scatter(x, y, color='green', s=1, alpha=0.8)
    ax1.set_xlim(-12.5, 12.5)
    ax1.set_ylim(-12.5, 12.5)
    ax1.set_aspect('equal')
    ax1.legend(["Bull's eye", "Yellow ring", "Blue ring", "Red ring", "White ring"])
    ax1.set_title("Simulated Dart Throws")

    ax2.hist(scores, bins=30, edgecolor='black')
    ax2.set_title(f"Distribution of Scores for {num_darts} Darts")
    ax2.set_xlabel("Score")
    ax2.set_ylabel("Frequency")

    plt.tight_layout()
    plt.show()

運行 1,000 局，每局五支飛鏢，並取落點座標標準差為 3 in.：

num_simulations = 1000
num_darts = 5
std_dev = 3

scores, ring_hits, all_throws = run_simulation(
    num_simulations, num_darts, std_dev
)

計算平均總分、各環命中機率與每環對單支飛鏢期望分數的貢獻：

mean_score = np.mean(scores)
total_throws = num_simulations * num_darts

print(f"Mean score for {num_darts} darts: {mean_score:.2f}")
print("\nProbability of hitting each ring:")
for ring, hits in ring_hits.items():
    prob = hits / total_throws
    expected_value = POINTS[ring] * prob
    print(f"{ring}: {prob:.4f} (Expected value: {expected_value:.2f})")

這個固定種子運行得到：

Mean score for 5 darts: 89.45

Bull's eye: 0.0566 (Expected value: 2.83)
Yellow ring: 0.2302 (Expected value: 5.75)
Blue ring: 0.4632 (Expected value: 6.95)
Red ring: 0.2220 (Expected value: 2.22)
White ring: 0.0274 (Expected value: 0.14)
Miss: 0.0006 (Expected value: 0.00)

找出期望貢獻最高的環：

best_ring = max(
    ring_hits.keys(),
    key=lambda x: POINTS[x] * ring_hits[x] / total_throws
)
print(f"\nRing with highest expected value: {best_ring}")

輸出為：

Ring with highest expected value: Blue ring

最後繪圖：

plot_dartboard(all_throws)

結果與討論

在 std_dev = 3 的指定二維常態模型與固定種子下，五支飛鏢平均總分為 $89.45$。各區結果為：

• 靶心：$0.0566$，期望貢獻 $2.83$
• 黃環：$0.2302$，期望貢獻 $5.75$
• 藍環：$0.4632$，期望貢獻 $6.95$
• 紅環：$0.2220$，期望貢獻 $2.22$
• 白環：$0.0274$，期望貢獻 $0.14$
• 脫靶：$0.0006$，期望貢獻 $0.00$

藍環的單支飛鏢期望分數貢獻最高，圖中的落點亦大多位於藍環。

這些數值不是飛鏢遊戲的一般規律，而是落點分佈、標準差、靶面尺寸及模擬種子共同產生的結果。若選手存在系統偏差、水平與垂直誤差不同，或投擲之間相依，結果會改變。

結語

本文以 Monte Carlo 方法模擬飛鏢得分，示範如何由一個明確機率模型估計平均總分、命中機率及期望貢獻。更有研究價值的下一步，是用實際落點資料估計分佈，再比較常態模型與帶偏差、重尾或學習效應的替代模型。