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Topic: Random movement probability problem (Read 1521 times) |
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marsh8472
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Random movement probability problem
« on: Sep 16th, 2012, 5:21am » |
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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?
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| « Last Edit: Sep 16th, 2012, 5:30am by marsh8472 » |
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marsh8472
Newbie
Posts: 27
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Re: Random movement probability problem
« Reply #1 on: Sep 16th, 2012, 6:19am » |
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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
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| « Last Edit: Sep 16th, 2012, 6:19am by marsh8472 » |
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towr
wu::riddles Moderator
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Re: Random movement probability problem
« Reply #2 on: Sep 16th, 2012, 6:53am » |
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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  L or d  -L
F(L,d,0) = 0 if -L < d < L
F(L,d,x) = 1/     0 F(L, |d + e  |, x-1 ) d  if -L < d < L and x > 0
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| « Last Edit: Sep 16th, 2012, 6:54am by towr » |
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