Question

一个立方体的每个面上都画有一条细窄条纹，从一条边的中点连到其对边的中点。每个面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圈的概率是多少？

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?

$\dfrac18$

$\dfrac{3}{16}$

$\dfrac14$

$\dfrac38$

$\dfrac12$

答案：B

知识点：基本概率独立事件对称性

难度评级：2090

解答：

每个面的条纹有 $2$ 种方向，所以共有 $2^6=64$ 种等可能配置。

一条环绕条纹会沿 $3$ 对相对面中的某一个方向，并经过 $4$ 个面；给定一个方向时，这四个面的条纹方向都被确定，所以概率为 $\left(\dfrac12\right)^4=\dfrac{1}{16}$ 。

三种方向互斥，所以总概率为 $3\cdot\dfrac{1}{16}=\dfrac{3}{16}$ 。

所以正确答案是 B。

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

An encircling stripe runs around one of the $3$ pairs of opposite faces. For a given band, the $4$ faces it crosses must each be oriented to continue it, a probability of $\left(\dfrac12\right)^4=\dfrac{1}{16}.$

The three bands are mutually exclusive, so the probability is $3\cdot\dfrac{1}{16}=\dfrac{3}{16}.$

Thus, the correct answer is B.

2009 AMC 10B 第 25 题

答案：B

难度评级：2090

解答：

其他年份的第 25 题