Question 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 $\dfrac{1}{2}$ and it disappears with probability $\dfrac{1}{2}.$
• For $n = 1,$ $2,$ or $3,$ if it is at $n,$ then it moves to $n+1$ with probability $\dfrac{1}{4},$ it moves to $n-1$ with probability $\dfrac{1}{4},$ and it disappears with probability $\dfrac{1}{2}.$
What is the probability that the frog reaches $4?$
• It starts at $0.$
• If it is at $0,$ then it moves to $1$ with probability $\dfrac{1}{2}$ and it disappears with probability $\dfrac{1}{2}.$
• For $n = 1,$ $2,$ or $3,$ if it is at $n,$ then it moves to $n+1$ with probability $\dfrac{1}{4},$ it moves to $n-1$ with probability $\dfrac{1}{4},$ and it disappears with probability $\dfrac{1}{2}.$
What is the probability that the frog reaches $4?$