# 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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## closed as not a real question by Asaf Karagila, azimut, Paul, Jim, user1729Apr 10 '13 at 8:46

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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 '13 at 20:39
can you explain why? –  MIMI Apr 9 '13 at 21:06
Start here: math.stackexchange.com/questions/536/… –  Byron Schmuland Apr 9 '13 at 21:35
–  Byron Schmuland Apr 9 '13 at 21:38