probability to reach a spesific point before other

Question

a frog can jump each time forward or backward (one step) with an equal probability of 0.5, the frog stands in point 0, what is the probability that the frog will reach to point 100 before point -1? tried to look at this answer but not sure that it is the same The probability of reaching a certain amount of money before going bust in a casino game.

Answer 1

Let $p_k$ be the probability of the frog's reaching $100$ before $-1$, if it is currently at $k$, where $1\leq k\leq100$. We have $$p_k=\cases{ 0,&$k=-1$\\ 1,&$k=100$\\ \frac{p_{k-1}}2+\frac{p_{k+1}}2,&otherwise}$$

It is easily seen that $$p_k=\frac{k+1}{101}$$ is the solution, so the desired probability is $$p_0=\frac1{101}$$

Answer 2

Your situation is closer to this version of gambler's ruin, where the random walk is symmetric. If you shift each position one unit to the right, then the frog starts at position ("state") $1$ and you want the probability that the frog hits state $101$ before hitting state $0$. This is the $p=\frac12$ version, so the desired probability equals $\frac{1}{101}$, since state $1$ is $\frac1{101}$ of the way between $0$ and $101$.