Question

Chloe 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 选择的数大于 Chloe 选择的数的概率是多少？

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

难度评级：1070

解答：

如果 Laurent 选到区间 $(2017, 4034]$ ，Chloe 就不可能有更大的数。

这意味着 Laurent 有 $\frac{1}{2}$ 的概率自动获胜。

否则，Laurent 选到区间 $[0, 2017]$ 。她比 Chloe 大的概率，与 Chloe 比 Laurent 大的概率相同。

因此 Laurent 有 $\frac{1}{2}$ 的概率取得更大的数（在实数区间中，由于区间无限，平局概率本质上是 $0$ ）。

Laurent 取得更大数的总概率为 $\dfrac{1}{2} \cdot 1 + \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{3}{4}.$

所以正确答案是 C。

If Laurent chooses a number in the interval $(2017, 4034],$ then there is no way that Chloé can have the greater number.

This means that Laurent has a $\frac{1}{2}$ chance of automatically winning.

Otherwise, Laurent chooses a number in the interval $[0, 2017].$ The probability that he gets a greater number than Chloé is the same as Chloé getting a greater number than Laurent.

This means that Laurent has a $\frac{1}{2}$ chance of getting a greater number (when working with real intervals, the probability of a tie is essentially $0$ due to the infinite size of the intervals).

Laurent's total chance of getting a greater number is $\dfrac{1}{2} \cdot 1 + \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{3}{4}.$

Thus, C is the correct answer.

2017 AMC 10A 第 15 题

答案：C

难度评级：1070

解答：

其他年份的第 15 题