Probability to reach $a>0$ before returning to the origin in a one dimensional random walk.

Question

Given a one dimensional random walk where I have $p$ probability to go forward and $q$ probability to go backwards.

I have to prove that, starting from the origin, the probability of reaching $a>0$ before returning to the origin is $p(1-q_1)$

Where

$$q_1=\frac{\left(\frac{q}{p}\right)^a-\left(\frac{q}{p}\right)}{\left(\frac{q}{p}\right)^a-1}$$

Answer 1

So first you have to go right, so that introduces a factor of $p$. After that, you need to hit $a$ before hitting $0$ starting at $1$. There is a standard way to compute the probability to do that, based on computing the probability to hit $a$ before hitting $0$ starting at every point in $0,1,\dots,a$. The idea is the total probability formula:

$$P(\tau_a<\tau_0 \mid X_0=k)=p P(\tau_a<\tau_0 \mid X_0=k+1) + q P(\tau_a<\tau_0 \mid X_0=k-1).$$

So if $u(k)=P(\tau_a<\tau_0 \mid X_0=k)$ then you have the equations

$$u(k)=pu(k+1)+qu(k-1)$$

for $k=1,2,\dots,a-1,u(0)=0,u(a)=1$. (Note that $u(0)$ is not the quantity that you want; you actually want $pu(1)$. This is an annoying thing about the distinction between "hitting" and "returning".)

This system can be solved; the solution is of the form

• $u(k)=c_1 + c_2 \left ( \frac{1-p}{p} \right )^k$ if $p \neq q$
• $u(k)=c_1 + c_2 k$ if $p=q$.

Using the BCs you can find $c_1,c_2$.

Answer 2

After what Ian's response.

Ian (https://math.stackexchange.com/users/83396/ian), Probability to reach $a>0$ before returning to the origin in a one dimensional random walk., URL (version: 2021-03-25): https://math.stackexchange.com/q/4076368

I thought that there was a paralelism between this problem and the gambler's ruin:

Let player A have a capital of $z$ and player B have a capital of $b$. If we define $a=z+b$. And the probability for the player A to win the game is A and the probability for player B to win the game is q. We know that the probability for player A to lose all his money is:

if $p\neq q$

$$q_z=\frac{\left(\frac{q}{p}\right)^a-\left(\frac{q}{p}\right)^z}{\left(\frac{q}{p}\right)^a-1}$$ if $p=q$ $$q_z=1-\frac{z}{a} $$

So in our model we need to redefine the $p_0$ because in the gambler's ruin it's $0$ (If you don't have money you can't gamble). But in our game we can actually go forward with a probability of $p%$.

In order to achieve the point $a$ you first have to start from the point $z=1$ for the probability to be defined in the gambler's ruin model. So, as Ian said, we need to go forward with a probability of $p$. Once we are in $z=1$, we can apply the gambler's ruin model so we have a probability of $p_1$ to reach $a$.

Finally we multiply all this two probabilities and we have the probability to reach $a$ starting from $0$. $$pp_1=p(1-q_1)$$