半失灵的口香糖机

Semi-Working Gum Machine

专题: Probability / 概率
难度: L3
来源: OpenQuant

题目详情

你和另外两个朋友从半功能的口香糖机中获取口香糖。本机将独立均匀地弹出一段长度为区间 $[0, 3]$ 的口香糖。在 3 根棍子中，你将收到最短的一根。你的口香糖棒的长度在 1 和 2 之间的概率是多少？

You and two other friends get gum from a semi-functioning gum machine. This machine will pop out a stick of gum with a length from the interval $[0, 3]$ independently and uniformly. Of the 3 sticks, you will receive the shortest one. What is the probability the length of your gum stick is between 1 and 2?

解析

如果最小长度在 1 到 2 之间，则所有棍子的长度必须至少为 1。这种情况发生的概率为 $\frac23^3$ 。然而，这包括所有口香糖棒都长于 2 的情况。减去这种情况，我们得到的概率为： $\frac23^3 - \frac13^3 = \boxed{\frac7{27}}$

import random

generate = lambda: random.uniform(0, 3)

valid_trial = 0
num_iters = 100_000
for i in range(num_iters):

    a = generate()
    b = generate()
    c = generate()

    smallest = min(a, b, c)
    if smallest > 1 and smallest < 2:
        valid_trial += 1

#expecting ~ 0.259
print(valid_trial/num_iters)

Original Explanation

If the minimum length is between 1 and 2, all sticks must be at least 1 long. This occurs with probability $\frac23^3$ . However, this includes the instance where all sticks of gum are longer than 2. Subtracting this case we get our probability to be: $\frac23^3 - \frac13^3 = \boxed{\frac7{27}}$

import random

generate = lambda: random.uniform(0, 3)

valid_trial = 0
num_iters = 100_000
for i in range(num_iters):

    a = generate()
    b = generate()
    c = generate()

    smallest = min(a, b, c)
    if smallest > 1 and smallest < 2:
        valid_trial += 1

#expecting ~ 0.259
print(valid_trial/num_iters)