涂红 10×10×10：随机小块红面数期望

Better in Red I

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

一个 10×10×10 的大立方体外表面全部涂成红色，然后切成 1000 个 1×1×1 小立方体。随机均匀选取一个小立方体。

问：该小立方体红色面的期望数量是多少？

A $10 \times 10 \times 10$ cube is painted red on the surface and then cut into $1000$ $1 \times 1 \times 1$ cubes and one is selected uniformly at random. Find the expected number of red faces on this cube.

解析

把所有小立方体的面汇总来看：

大立方体的外表面包含 $6\cdot 10^2=600$ 个被涂红的小面。

随机选取一个小立方体，相当于随机抽取其 6 个面中的每一个是否被涂红；红面总数的期望等于“总红面数 / 总小立方体数”。

因此期望为

\frac{600}{1000}=\frac{3}{5}.

Original Explanation

Label the faces of each cube $1-6$ , and then let $I_i$ be the indicator that side $i$ of the cube that is drawn is colored red. Then $T = I_1 + \dots + I_6$ gives the total number of red sides of the cube. By linearity of expectation and the fact that this is the cube so all the sides are exchangeable, $\mathbb{E}[T] = 6\mathbb{E}[I_1]$ . $\mathbb{E}[I_1]$ is just the probability that side $1$ is colored red. We know that $10^2$ of the cubes will have side $1$ colored red, as each side is colored red on one face of the big cube, which is $10^2$ little cubes. Therefore, the probability side 1 is red is $\dfrac{10^2}{10^3} = \dfrac{1}{10}$ . Therefore, $\mathbb{E}[T] = \dfrac{3}{5}$ by substituting back in.