10个小球，随机扔到12个盒子里，求恰好有10个盒子都为空的概率。题目要求使用暴力模拟10万次来算一下。 原始题目是有人去字节面试，这个是面试题目的第一题。原文在此：https://www.nowcoder.com/discuss/353156239813189632

题目本身不难，只是因为概率太低，使用10万次模拟往往一次结果也没有。最终计算结果，这个概率大约为100万分之一，详细解释上文也给出了，所以最好使用100万次的模拟。

概率计算： C(12, 2) * (2^10 -2) / 12^10 ~= 0.000001089

这个概率极低，这个事情最有意思的一点是，如果不是问恰好10个盒子都是空，而是9个盒子为空或8个为空，那这个概率就会增加很多，这也跟我们实际经验相符。

恰好9个盒子为空的概率: C(12, 3) * (3^10 - C(3,1) - C(3,2) * (2^10 - 2)) / 12^10 ~= 0.0001989

恰好8个盒子为空的概率(这个式子有点复杂，里面包含了上面式子的一部分，我之前列式错了，得到的结果为0.008，跟程序模拟的结果怎么也对不上，后来发现式子列的有问题): C(12, 4) * (4^10 - C(4,1) - C(4,2) * (2^10 -2) - C(4,3) * (3^10 - C(3,1) - C(3,2) * (2^10 - 2))) / 12^10 ~= 0.006544

import random

NUM_OF_BOXES = 12
NUM_OF_BALLS = 10

def try_once(blank_box_num):
    boxes = [False for _ in range(NUM_OF_BOXES)] 
    for _ in range(NUM_OF_BALLS): 
        picked_box_num = random.randint(0, NUM_OF_BOXES -1) 
        boxes[picked_box_num] = True
    return boxes.count(False) == blank_box_num

def try_many(times, blank_box_num):
    bingo = 0
    for _ in range(times):
        if(try_once(blank_box_num)):
            bingo = bingo + 1
    print(f"The ratio of there are only \
        {blank_box_num} boxes empty: {bingo}/{times}")

# Test
for _ in range(5):
    try_many(1000000, 10)

for _ in range(5):
    try_many(1000000, 9)

for _ in range(5):
    try_many(1000000, 8)

程序运行结果如下，跟计算的概率相当匹配：

The ratio of there are only 10 boxes empty: 2/1000000
The ratio of there are only 10 boxes empty: 1/1000000
The ratio of there are only 10 boxes empty: 0/1000000
The ratio of there are only 10 boxes empty: 0/1000000
The ratio of there are only 10 boxes empty: 0/1000000
The ratio of there are only 9 boxes empty: 194/1000000
The ratio of there are only 9 boxes empty: 163/1000000
The ratio of there are only 9 boxes empty: 194/1000000
The ratio of there are only 9 boxes empty: 200/1000000
The ratio of there are only 9 boxes empty: 197/1000000
The ratio of there are only 8 boxes empty: 6536/1000000
The ratio of there are only 8 boxes empty: 6516/1000000
The ratio of there are only 8 boxes empty: 6551/1000000
The ratio of there are only 8 boxes empty: 6511/1000000
The ratio of there are only 8 boxes empty: 6560/1000000