Question

一对标准 $6$ 面公平骰子掷一次。掷出的点数和决定一个圆的直径。圆的面积的数值小于该圆周长的数值的概率是多少？

A pair of standard $6$ -sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

$\dfrac{1}{36}$

$\dfrac{1}{12}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{5}{18}$

答案：B

知识点：骰子（概率）圆不等式

难度评级：1370

解答：

若直径为 $d$ ，面积 $\lt$ 周长意味着 $\dfrac{\pi d^2}{4} \lt \pi d$ ，即 $d \lt 4$ 。因为 $d \ge 2$ ，所以点数和必须是 $2$ 或 $3$ 。

点数和为 $2$ 的概率是 $\dfrac{1}{36}$ ，点数和为 $3$ 的概率是 $\dfrac{2}{36}$ ，合计 $\dfrac{3}{36} = \dfrac{1}{12}$ 。

因此，正确答案是 B。

For diameter $d,$ area $\lt$ circumference means $\dfrac{\pi d^2}{4} \lt \pi d,$ i.e. $d \lt 4.$ Since $d \ge 2,$ this needs a sum of $2$ or $3.$

A sum of $2$ has probability $\dfrac{1}{36}$ and a sum of $3$ has probability $\dfrac{2}{36},$ totaling $\dfrac{3}{36} = \dfrac{1}{12}.$

Thus, the correct answer is B.

2011 AMC 12A 第 10 题

答案：B

难度评级：1370

解答：

其他年份的第 10 题