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January 26th, 2011, 08:33 AM   #1
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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?
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January 26th, 2011, 08:36 AM   #2
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Re: Random Walk question

I should probably request a transfer to College Mathematics. I missed the difference between the sections in my anxiety.
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January 26th, 2011, 08:46 AM   #3
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Re: Random Walk question

The following link will lead to material that may be of interest:

http://en.wikipedia.org/wiki/Infinite_monkey_theorem
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January 26th, 2011, 11:40 AM   #4
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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 ((x-y+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) = (1-P(0)) * (1-P(1)) * (1-P(2)) * ... * (1-P(t-1)) * 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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