I know that this may seem easy but I have no clue where to start (if possible could you answer this in the simplest way possible)?
Consider a person who is at the position $x=0$ on the $x$-axis at time $0$. At time $t=1$ he moves to $x = 1$ or $x = −1$ with probability $1/2$. After $t$ seconds if he is in position $x$, he will move to $x + 1$ or $x − 1$ with probability $1/2$. Discuss the following questions:
- What is the probability that he will be back at position $0$?
- Given a fixed point $X$, what is the probability that he reaches $X$, if we do not mind how long it takes?
- Discuss the same problem in two and three dimensional space. In $2$D, he moves north, south, west or east, each one with the probability $1/4$. In $3$D there are $6$ directions he might take, each one with probability $1/6$.