AMC 12 2025b Problem 20

A frog hops along the number line according to the following rules:

• It starts at $0$.
• If it is at $0$, then it moves to $1$ with probability $\frac 12$ and disappears with probability $\frac 12$.
• For $n=1,2,3$, if it is at $n$, then it moves to $n+1$ with probability $\frac 14$, to $n-1$ with probability $\frac 14$, and disappears with probability $\frac 12.{}$

What is the probability that the frog reaches $4$?

$\textbf{(A) } \frac1{101} \textbf{(B) } \frac 1{100} \textbf{(C) } \frac1{99} \textbf{(D) } \frac1{98} \textbf{(E) } \frac1{97}$