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        riddles >> medium >> Random movement probability problem
        
(Message started by: marsh8472 on Sep 16^th, 2012, 5:21am)

Title: Random movement probability problem
Post by marsh8472 on Sep 16^th, 2012, 5:21am

http://www.thedogtrainingsecret.com/s/3M_YReISLUOdzOFOpj1Wbg/dogpullingleash.png

A dog is put in the backyard on a leash that is L units long. Every second a dog moves 1 unit in a randomly picked direction (any real number from 0 to 360 degrees). What is the probability that this dog will reach the end of its leash at least once within the first X seconds?

Title: Re: Random movement probability problem
Post by marsh8472 on Sep 16^th, 2012, 6:19am

There's probably recursion and integration involved with this

Coordinates after X seconds =
(0,0), if X=0
Coordinates after X-1 seconds + (cos(random), sin(random)), if X>0

Title: Re: Random movement probability problem
Post by towr on Sep 16^th, 2012, 6:53am

I think the following recursion should describe the solution, but it doesn't seem very useful for anything other than finding numerical solutions to specific cases.
F(L,d,x) = 1 if d http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ge.gif L or d http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif -L
F(L,d,0) = 0 if -L < d < L
F(L,d,x) = ¹/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sub2.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subpi.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sup2.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/suppi.gif₀ F(L, |d + ehttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supi.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/suptheta.gif|, x-1 ) d http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif if -L < d < L and x > 0