2025 AMC 10B Problem 24

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 \tfrac12 and it disappears with probability \tfrac12. For n = 1, 2, or 3, if it is at n, then it moves to n + 1 with probability \tfrac14, it moves to n - 1 with probability \tfrac14, and it disappears with probability \tfrac12.

What is the probability that the frog reaches 4?

\textbf{(A)} \dfrac{1}{101} \qquad \textbf{(B)} \dfrac{1}{100} \qquad \textbf{(C)} \dfrac{1}{99} \qquad \textbf{(D)} \dfrac{1}{98} \qquad \textbf{(E)} \dfrac{1}{97}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.