涂色 5×5×5 立方体：随机小块掷出红面朝上的概率

等价于在所有小立方体的所有面中，随机选取一个面看它是否为红。

• 总涂色面数：大立方体外表面面积为 $6\cdot 5^2=150$。
• 小立方体总面数：$125\cdot 6=750$。

因此概率为

$$\frac{150}{750}=\frac{1}{5}.$$

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Original Explanation

Let's solve the generic case for a $(n \times n \times n)$ cube. In particular, there are $6n^2$ painted sides, as each side has $n^2$ faces painted. There are $6n^3$ total faces in the cube, as each of the $n^3$ cubes has $6$ faces. Therefore, the probability that we obtained a painted side when we roll the selected $(1 \times 1 \times 1)$ cube is $\frac{6n^2}{6n^3} = \frac{1}{n}$. In this case, $n = 5$, so our answer is $\frac{1}{5}$.