2025 AMC 10B Problem 24

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

$\qquad\bullet$ It starts at $0$ .

$\qquad\bullet$ If it is at $0$ , then it moves to $1$ with probability $\tfrac{1}{2}$ and it disappears with probability $\tfrac{1}{2}$ .

$\qquad\bullet$ For $n = 1, 2,$ or $3,$ if it is at $n,$ then it moves to $n+1$ with probability $\tfrac{1}{4},$ it moves to $n-1$ with probability $\tfrac{1}{4},$ and it disappears with $\qquad\phantom{\bullet}$ probability $\tfrac{1}{2}$ .

What is the probability that the frog reaches ${4\,?}$