Question

Chloé 从区间 $[0,2017]$ 中均匀随机选择一个实数。Laurent 独立地从区间 $[0,4034]$ 中均匀随机选择一个实数。Laurent 选到的数大于 Chloé 选到的数的概率是多少？

Chloé chooses a real number uniformly at random from the interval $[0,2017].$ Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034].$ What is the probability that Laurent's number is greater than Chloé's number?

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{5}{6}$

$\dfrac{7}{8}$

答案：C

知识点：几何概率对称性对立事件概率

难度评级：1560

解答：

Laurent 的数有概率 $\dfrac{1}{2}$ 落在 $[2017,4034]$ ，这会大于 Chloé 可能选择的任何数，因此他必胜。

另有概率 $\dfrac{1}{2}$ ，Laurent 的数落在 $[0,2017]$ ，与 Chloé 的区间相同；由对称性，他较大的概率是一半。总概率为 $\dfrac{1}{2}\cdot1+\dfrac{1}{2}\cdot\dfrac{1}{2}=\dfrac{3}{4}.$

所以正确答案是 C。

With probability $\dfrac{1}{2},$ Laurent's number lies in $[2017,4034],$ which exceeds any number Chloé could choose, so he wins for certain.

With the other probability $\dfrac{1}{2},$ Laurent's number lies in $[0,2017],$ matching Chloé's interval; by symmetry he is larger half the time. The total probability is $\dfrac{1}{2}\cdot1+\dfrac{1}{2}\cdot\dfrac{1}{2}=\dfrac{3}{4}.$

Thus, the correct answer is C.

2017 AMC 12A 第 10 题

答案：C

难度评级：1560

解答：

其他年份的第 10 题