情侣期望数

Expected Couples

专题: Probability / 概率
难度: L3
来源: OpenQuant

题目详情

来自四对已婚夫妇的八个人被随机分配到圆桌周围的座位。预计坐在一起的情侣数量是多少？

Eight people from four married couples are randomly assigned to seats around a round table. What is the expected number of couples that are seated next to each other?

解析

让我们来看看一位特定的女性。该女士旁边有 $2$ 座位，她的丈夫随机坐在 $7$ 剩余座位之一上。因此，女人坐在她丈夫旁边的概率是 $\frac27$ 。由于座位是随机分配的，因此没有女性坐在丈夫旁边的可能性不同，因此坐在丈夫旁边的女性（或夫妻总数）的预期数量是： $4\cdot\frac27 = \boxed{\frac87}$

import random

couple_counts = []
num_iters = 100_000
for i in range(num_iters):

    #where 1/2, 3/4, 5/6, and 7/8 are couples
    people = [1, 2, 3, 4, 5, 6, 7, 8]
    random.shuffle(people)
    people.append(people[0])
    num_couples = 0

    for p in range(8):

        if people[p] % 2 == 0:
            if people[p] == people[p+1] + 1:
                num_couples += 1
        else:
            if people[p] == people[p+1] -1:
                num_couples += 1

    couple_counts.append(num_couples)

#expecting 8/7 ~ 1.1429
print(sum(couple_counts) / num_iters)

Original Explanation

Let's take a look at a given woman. There are $2$ seats adjacent to this woman and her husband is randomly seated in one of the $7$ remaining seats. Thus the probability the woman sits next to her husband is $\frac27$ . Since seating is assigned at random, no woman has a different likelihood to be seated next to her husband and so the expected number of woman that sit next to their husband (or total couples is): $4\cdot\frac27 = \boxed{\frac87}$

import random

couple_counts = []
num_iters = 100_000
for i in range(num_iters):

    #where 1/2, 3/4, 5/6, and 7/8 are couples
    people = [1, 2, 3, 4, 5, 6, 7, 8]
    random.shuffle(people)
    people.append(people[0])
    num_couples = 0

    for p in range(8):

        if people[p] % 2 == 0:
            if people[p] == people[p+1] + 1:
                num_couples += 1
        else:
            if people[p] == people[p+1] -1:
                num_couples += 1

    couple_counts.append(num_couples)

#expecting 8/7 ~ 1.1429
print(sum(couple_counts) / num_iters)