Question

一个立方体的每个面都独立地以概率 $\tfrac12$ 涂成红色或蓝色。这个涂色立方体能被放在水平面上，使得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：正方体分类讨论基本概率

难度评级：1890

小提示：

共有 $2^6 = 64$ 种等可能的涂色。

There are $2^6 = 64$ equally likely colorings

大提示：

统计全部 $6$ 面同色、恰好 $5$ 面同色，或 $4\text{-}2$ 分布且少数颜色在相对两面的情况。

Count arrangements with all $6,$ exactly $5,$ or a $4\text{-}2$ split on opposite faces

解答：

共有 $2^6 = 64$ 种涂色。存在合适摆放方式的情况包括：六个面全同色（ $2$ 种），恰好五个面同色（ $2 \cdot 6 = 12$ 种），或四个面为一种颜色且另外两个面为另一种颜色并位于相对面（ $2 \cdot 3 = 6$ 种）。共有 $2 + 12 + 6 = 20$ 种有利涂色，所以概率为 $\dfrac{20}{64} = \dfrac{5}{16}$ 。

因此正确答案是 B。

There are $2^6 = 64$ colorings. A suitable orientation exists when all six faces are one color ( $2$ ways), exactly five faces are one color ( $2 \cdot 6 = 12$ ways), or four faces are one color with the other color on a pair of opposite faces ( $2 \cdot 3 = 6$ ways). That is $2 + 12 + 6 = 20$ favorable colorings, so the probability is $\dfrac{20}{64} = \dfrac{5}{16}.$

Thus, the correct answer is B.

2004 AMC 12B 第 20 题

答案：B

难度评级：1890

小提示：

大提示：

解答：

其他年份的第 20 题