Question

立方体的每个面上都画有一条窄条纹，从一条边的中点连到其对边的中点。每个面上选择哪一对对边是随机且相互独立的。出现一条连续条纹环绕立方体一周的概率是多少？

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?

$\dfrac{1}{8}$

$\dfrac{3}{16}$

$\dfrac{1}{4}$

$\dfrac{3}{8}$

$\dfrac{1}{2}$

答案：B

知识点：基本概率正方体分类讨论

难度评级：1890

小提示：

每个面的条纹有 $2$ 个方向，所以共有 $2^6 = 64$ 种等可能配置。

Each face’s stripe has $2$ orientations, giving $2^6 = 64$ equally likely configurations

大提示：

环绕条纹会沿着 $3$ 对相对面中的某一对绕行；数出能形成环的配置。

An encircling stripe runs around one of $3$ pairs of opposite faces; count the configurations forming a loop

解答：

$6$ 个面中的每个面都有 $2$ 个等可能的条纹方向，所以共有 $2^6 = 64$ 种配置。

环绕条纹沿着 $3$ 对相对面中的某一对绕行。固定这样一条带，条纹经过的四个面必须对齐，概率为 $\left(\dfrac{1}{2}\right)^4 = \dfrac{1}{16}$ ，其余两个面任意。 $3$ 条可能的带互不重叠，所以概率为 $3 \cdot \dfrac{1}{16} = \dfrac{3}{16}$ 。

所以正确答案是 B。

Each of the $6$ faces has $2$ equally likely stripe orientations, for $2^6 = 64$ configurations.

An encircling stripe runs around one of the $3$ pairs of opposite faces. Fixing such a band, the four faces it passes through must be aligned, with probability $\left(\dfrac{1}{2}\right)^4 = \dfrac{1}{16},$ while the two remaining faces are free. The $3$ possible bands are disjoint events, so the probability is $3 \cdot \dfrac{1}{16} = \dfrac{3}{16}.$

Thus, the correct answer is B.

2009 AMC 12B 第 17 题

答案：B

难度评级：1890

小提示：

大提示：

解答：

其他年份的第 17 题