Question

一只蜜蜂在三维空间中移动。掷一个公平六面骰，其面标为 $A^+, A^-, B^+, B^-, C^+$ 、 $C^-$ 。若蜜蜂位于点 $(a, b, c)$ ，骰子显示 $A^+$ 时蜜蜂移动到 $(a + 1, b, c)$ ，显示 $A^-$ 时移动到 $(a - 1, b, c)$ 。其余四种结果作类似移动。

假设蜜蜂从 $(0, 0, 0)$ 出发，骰子掷四次。蜜蜂走过某个单位立方体的四条不同边的概率是多少？

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+,$ and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c).$ If the die shows $A^+,$ then the bee moves to the point $(a + 1, b, c),$ and if the die shows $A^-,$ then the bee moves to the point $(a - 1, b, c).$ Analogous moves are made with the other four outcomes.

Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?

$\dfrac{1}{54}$

$\dfrac{7}{54}$

$\dfrac{1}{6}$

$\dfrac{5}{18}$

$\dfrac{2}{5}$

答案：B

知识点：基本概率正方体分类讨论

难度评级：2380

解答：

每次掷骰都会让蜜蜂沿 $\pm x,\pm y$ 或 $\pm z$ 方向移动一个单位，所以共有 $6^4=1296$ 个等可能移动序列。有效路径分成两类。第一类走过某个正方形面的四条边：坐标方向对有 $3$ 种选择，两个方向的正负号有 $4$ 种选择，第一个方向有 $2$ 种选择，共 $24$ 条路径。第二类用到三个坐标方向，其中一个方向使用两次。重复方向有 $3$ 种选择；它在四步方向序列中的两个位置不能相邻，这两个位置有 $3$ 种选法，其余两个方向的顺序有 $2$ 种。对每个这样的方向序列，恰有 $2^3=8$ 种正负号选择能使四条不同的边位于同一个立方体上。因此第二类有 $3\cdot3\cdot2\cdot8=144$ 条路径。总共有 $24+144=168$ 个有利序列，所以概率为 $168/1296=7/54$ ，正确答案是 B。

Every roll moves the bee one unit along $\pm x,\pm y,$ or $\pm z,$ so there are $6^4=1296$ equally likely sequences. There are two types of valid paths. A path around one square face has $3$ choices of coordinate plane, $4$ choices for the signs of its two axes, and $2$ choices for which axis is used first, giving $24.$ Otherwise all three coordinate directions are used, with one repeated: choose that axis in $3$ ways, choose its two nonadjacent positions in $3$ ways, order the other two axes in $2$ ways, and choose their three initial signs in $2^3=8$ ways. (The second step on the repeated axis must have the opposite sign.) This gives $3\cdot3\cdot2\cdot8=144.$ Hence there are $24+144=168$ favorable sequences, and the probability is $168/1296=7/54.$ Therefore, the answer is B.

2024 AMC 10A 第 24 题

答案：B

难度评级：2380

解答：

其他年份的第 24 题