2004 AMC 12B 第 20 题

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：

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：

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：

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