涂红 10×20×30：随机小块红面数期望

总红色小面数等于长方体表面积：

$$2(10\cdot 20+20\cdot 30+10\cdot 30)=2(200+600+300)=2200.$$

共有 6000 个小立方体，因此随机小立方体红面数期望为

$$\frac{2200}{6000}=\frac{11}{30}.$$

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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. We need to be careful here, as the indicators are not exchangeable. Instead, two of the indicators will correspond to a $10 \times 20$ side, two to a $20 \times 30$ side, and two to a $10 \times 30$ side. Therefore, there are 3 subsets of indicators that are exchangeable, so we can reduce it to an expectation involving 3 indicators and multiply it by 2, so $\mathbb{E}[T] = 2(\mathbb{E}[I_1] + \mathbb{E}[I_2] + \mathbb{E}[I_3])$, where we take $I_1$ to indicate a $10 \times 20$ side, $I_2$ to indicate a $20 \times 30$ side, and $I_3$ to indicate a $10 \times 30$ side.

Then, to evaluate these expectations, we just need to find the probability that the side indicated is red on our cube. For $I_1$, that is $10 \cdot 20 = 200$ cubes. For $I_2$, that is $20 \cdot 30 = 600$ cubes. For $I_3$, that is $10 \cdot 30 = 300$ cubes. All of these need to be divided by $6000$ as that is the volume of the entire prism. Therefore, $\mathbb{E}[T] = \dfrac{2(200 + 300 + 600)}{6000} = \dfrac{11}{30}$.