# 验证: 抛硬币 1000 次
import numpy as np
np.random.seed(42)
flips = np.random.choice(["H", "T"], 1000)
p_head = (flips == "H").mean()
print(f"1000 次抛硬币: 正面频率 = {p_head:.3f} (理论 0.500)")
# 10000 次: 更接近 0.5
flips_10k = np.random.choice(["H", "T"], 10000)
print(f"10000 次: 正面频率 = {(flips_10k == 'H').mean():.4f}")

# 例: 10 个灯泡, 3 个坏的, 抽 2 个 (不放回), 至少 1 个坏的概率
from math import comb
n_total = comb(10, 2)              # 总抽法 C(10,2) = 45
n_no_bad = comb(7, 2)              # 都没坏 C(7,2) = 21
p_at_least_one_bad = 1 - n_no_bad / n_total
print(f"P(至少 1 坏) = 1 - 21/45 = {p_at_least_one_bad:.3f}")

# 例: 抽 5 张扑克, 都是红桃的概率 (不放回)
p_heart = 13/52
# 第 1 张: 13/52, 第 2 张: 12/51, ...
p_all_hearts = 1
for i in range(5):
    p_all_hearts *= (13 - i) / (52 - i)
print(f"P(5 张都红桃) = {p_all_hearts:.6f}")  # ≈ 0.0005

from math import factorial
# 5! = 120
print(factorial(5))

from math import perm
# 5 个里选 3 个排列: P(5,3) = 60
print(perm(5, 3))

from math import comb
# 5 个里选 3 个组合: C(5,3) = 10
print(comb(5, 3))

import numpy as np

def monty_hall(n_trials=100_000, switch=True):
    """模拟 Monty Hall 游戏"""
    wins = 0
    for _ in range(n_trials):
        # 随机放车
        car = np.random.randint(0, 3)             # 0, 1, 2
        # 你随机选
        choice = np.random.randint(0, 3)
        # 主持人开 1 扇 (不是你选的, 且不是车)
        available = [d for d in range(3) if d != choice and d != car]
        host_opens = np.random.choice(available)
        # 如果换, 你选剩下那扇
        if switch:
            choice = [d for d in range(3) if d not in (choice, host_opens)][0]
        if choice == car:
            wins += 1
    return wins / n_trials

print(f"不换: 胜率 = {monty_hall(switch=False):.4f} (理论 1/3)")
print(f"换:   胜率 = {monty_hall(switch=True):.4f} (理论 2/3)")

不换: 胜率 = 0.3335 (理论 1/3 = 0.3333)
换:   胜率 = 0.6668 (理论 2/3 = 0.6667)