Question

一个立方体的每个面独立地以概率 $\tfrac12$ 涂成红色或蓝色。这个涂色立方体可以放在水平面上，使四个竖直面颜色全相同的概率是多少？

Each face of a cube is painted either red or blue, each with probability $\tfrac12.$ The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

$\dfrac{1}{4}$

$\dfrac{5}{16}$

$\dfrac{3}{8}$

$\dfrac{7}{16}$

$\dfrac{1}{2}$

答案：B

知识点：基本概率分类讨论正方体

难度评级：1990

解答：

固定立方体方向，共有 $2^6 = 64$ 种涂色。

可行情况包括：六面同色，有 $2$ 种；恰好五面同色，有 $\binom{6}{5} \cdot 2 = 12$ 种；以及三组相对面中选一组作上下底面，其余四个竖直面同色，有 $3$ 组相对面、 $2$ 种颜色，共 $6$ 种。

总共有 $2 + 12 + 6 = 20$ 种，所以概率为 $\dfrac{20}{64} = \dfrac{5}{16}$ 。

所以正确答案是 B。

Fixing the orientation, there are $2^6 = 64$ colorings.

A coloring works if all six faces match ( $2$ ways), exactly five match ( $\binom{6}{5} \cdot 2 = 12$ ways), or four faces share a color with the remaining pair being opposite faces of the other color ( $3$ opposite pairs, $2$ colors, giving $6$ ways).

The total is $2 + 12 + 6 = 20,$ so the probability is $\dfrac{20}{64} = \dfrac{5}{16}.$

Thus, the correct answer is B.

2004 AMC 10B 第 23 题

答案：B

难度评级：1990

解答：

其他年份的第 23 题