Question

蜘蛛 Charlotte 在图中形状像 $5$ 角星的蛛网上随机行走。蛛网有 $5$ 个外顶点和 $5$ 个内顶点。每次到达一个顶点时，她都会随机选择一个相邻顶点并走过去。她从一个外顶点出发，走 $3$ 步后在外顶点的概率是多少？

Charlotte the spider is walking along a web shaped like a $5$ -pointed star, shown in the figure below. The web has $5$ outer points and $5$ inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes $3$ moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{2}{5}$

$\dfrac{1}{2}$

$\dfrac{3}{5}$

答案：B

知识点：随机游走基本概率

难度评级：1590

视频讲解：

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文字解答：

从外顶点出发，Charlotte 第一步一定到内顶点。从内顶点出发， $4$ 个相邻点中有 $2$ 个是外顶点，所以走到外顶点的概率是 $1/2$ 。一步后她在内顶点。两步后她有 $1/2$ 的概率在外顶点、 $1/2$ 的概率在内顶点。第三步后在外顶点，只能来自两步后在内顶点的情况，这一步概率为 $1/2$ ，所以最终概率为 $\frac12\cdot\frac12=\frac14$ 。

From an outer point, Charlotte must move to an inner point. From an inner point, $2$ of the $4$ neighboring points are outer, so the probability of moving to an outer point is $1/2$ . After one move she is inner. After two moves she is outer with probability $1/2$ and inner with probability $1/2$ . After the third move, only the inner case can move to outer, with probability $1/2$ , so the final probability is $\frac12\cdot\frac12=\frac14$ .

2026 AMC 8 第 21 题

答案：B

难度评级：1590

视频讲解：

文字解答：

其他年份的第 21 题