# Random walks in $1$, $2$ and $3$ dimensions [closed]

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:

1. What is the probability that he will be back at position $0$?
2. Given a fixed point $X$, what is the probability that he reaches $X$, if we do not mind how long it takes?
3. 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 $1/6$.
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Nice question, doesn't seem that basic to me for higher dimensions. For 1 dimensions the answer is it will return with probability $1$. –  muzzlator Apr 9 at 20:39
can you explain why? –  MIMI Apr 9 at 21:06
–  Byron Schmuland Apr 9 at 21:38