Time Series Asset Probability

Let’s say the price of an asset behaves as follows:

The asset price starts from 0 At any time t, the price is equally likely to go up or down by 1 i.e. Price(t+1) can only be Price(t) + 1 or Price(t) – 1, and both of those are equally likely However, if the price goes to 10, it has to go back to 9 at the next instant. If the price goes to -10, it has to go back to -9 at the next instant. That is, if Price(t) = 10, Price(t+1) has to be 9. If Price(t) = -10, Price(t+1) has to be -9.

What is the Fair Value of the Asset(Expected Value)?

Calculate Probability of Asset being each of the Values(i.e -10 to +10)?

Prove that ExpectedValue(t)=ExpectedValue(t+1)?

Answer contains this Question

If current price of the stock is 0, what is the probability that stock price goes to 2 before it goes to -4?

Note:Please try to explain by equations and not by intuition because of the symmetries involved.

Please see the attachment for a visual representation

Also if you can refer resources to learn these type of problems, it would be great!