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January 26th, 2011, 08:33 AM  #1 
Newbie Joined: Jun 2010 Posts: 11 Thanks: 0  Random Walk question
I've had a question about random walks ravaging me for the last week or so... well, a couple actually. The walk takes place on a square lattice on an infinite plane. Each step, the walk can move 1 grid unit on either the X or Y axis. I have two points, origin and some point (x,y). What is the probability that after t steps the walk is at (x,y) for the first time? What is the average value of t when the walk reaches (x,y) for the first time? I just don't understand probability well enough to formulate this, and as such I have been resorting to graphing the possibilities out. This gets exponentially more tedious and prone to error with each step of t. Any help? 
January 26th, 2011, 08:36 AM  #2 
Newbie Joined: Jun 2010 Posts: 11 Thanks: 0  Re: Random Walk question
I should probably request a transfer to College Mathematics. I missed the difference between the sections in my anxiety.

January 26th, 2011, 08:46 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  Re: Random Walk question
The following link will lead to material that may be of interest: http://en.wikipedia.org/wiki/Infinite_monkey_theorem 
January 26th, 2011, 11:40 AM  #4 
Newbie Joined: Jun 2010 Posts: 11 Thanks: 0  Re: Random Walk question
The probability P that the walk is at a point (x,y) at time t is given by P(t) = (t choose ((x+y+t)/2)) * (t choose ((xy+t)/2)) / 4^t The probability K that at time t the walk is at point (x,y) for the first time is given by: K(t) = (1P(0)) * (1P(1)) * (1P(2)) * ... * (1P(t1)) * P(t) Now I need to simplify this to cope with the exponential growth of significant factors, since the most likely length is ~x^2+y^2 (consistent with the mean square distance from origin of a random walk being ~sqrt(t)) 

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