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 (revisiting points is allowed). What is the probability she is now at one of the inner points of the star ?

A. $\frac{1}{4}$

B. $\frac{2}{5}$

C. $\frac{1}{2}$

D. $\frac{3}{5}$

E. $\frac{3}{4}$

1. 2026 AMC 8 官方解析

$P_{\text{外} \to \text{内}} = 1$ , $P_{\text{内} \to \text{外}} = \frac{1}{2}$ , $P_{\text{内} \to \text{内}} = \frac{1}{2}$

$\text{外} \xrightarrow{\quad 1\quad } \text{内} \begin{cases} \xrightarrow{\frac{1}{2}} \text{外} \xrightarrow{\quad 1\quad } \text{内} \\ \xrightarrow{\frac{1}{2}} \text{内} \xrightarrow{\quad \frac{1}{2}\quad } \text{内} \end{cases}$

$P = 1 \times \frac{1}{2} \times 1 + 1 \times \frac{1}{2} \times \frac{1}{2} = \frac{3}{4}$