2021 AMC 10B Spring 第 18 题

Question

反复掷一枚公平的

6

面骰，直到第一次出现奇数为止。在第一次出现奇数之前，每个偶数都至少出现一次的概率是多少？

A fair

6

-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?

\dfrac{1}{120}

\dfrac{1}{32}

\dfrac{1}{20}

\dfrac{3}{20}

\dfrac{1}{6}

答案：C

知识点：骰子（概率）基本概率

难度评级：1220

小提示：

暂时忽略重复掷出的结果，只看第一次出现的新点数类型。

Ignore repeated rolls until a new face type appears

大提示：

前三个不同的结果必须先是三个偶数，然后才出现任何奇数。

The first three distinct outcomes must be the three even numbers before any odd number appears

视频讲解：

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文字解答：

第一个不同的点数是偶数的概率为

\frac 36

。

在已经出现一个偶数后，第二个不同点数仍为偶数的概率为

\frac 25

。

在已经出现两个不同偶数后，第三个不同点数仍为偶数的概率为

\frac 14

。

因此所求概率为

\dfrac{3\cdot 2\cdot 1}{6\cdot 5\cdot 4} = \dfrac 1{20}\text{。}

所以答案是 C。

The probability that the first number is even is

\frac 36.

The probability that the second distinct number is even is

\frac 25.

The probability that the third distinct number is even is

\frac 14.

The combined probability is

\dfrac{3\cdot 2\cdot 1}{6\cdot 5\cdot 4} = \dfrac 1{20}.

Thus, the answer is C .