2017 AMC 10A 第 15 题

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}

把 Laurent 的区间分成下半段和上半段。

Break Laurent’s interval into the lower half and upper half

在下半段中，由对称性判断谁的数更大。

In the lower half, symmetry decides which chosen number is larger

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]

。此时 Laurent 选到更大数的概率，与 Chloe 选到更大数的概率相同。

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}\text{。}

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.

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