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Topic: Caterpillar on a Grid (Read 277 times) |
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ThudnBlunder
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A caterpillar is placed on a 2 x 3 grid at the position marked X in the figure below. It moves along the edges of the grid as follows: at an intersection it randomly chooses one of the edges leading from that intersection and moves along it, arriving at the next intersection one "step" later. (It is allowed to immediately double back on its path.) The points marked A and B are exits. When the caterpillar gets to one of them it leaves the grid.
1) Is the caterpillar more likely to exit through point A or point B?
2) On average, how long will it take the caterpillar to exit the grid (either through point A or point B)?
3) If the caterpillar is not allowed to immediately double back on its path, how does this affect the answers to 2) and 3)?
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towr
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Re: Caterpillar on a Grid
« Reply #1 on: Nov 23rd, 2003, 9:42am » |
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1) ::point A is slightly more likely, about 0.50413 vs 0.49587. I got that result by creating the transition matrix and raising it to the power 32768, after which I applied it to the starting state vector.::
2) ::349/35, from solving the equalities you get by observing the av. path-length starting at some point (except a and b) equals 1 + the av. path-length starting at the neighbours::
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| « Last Edit: Nov 23rd, 2003, 11:10am by towr » |
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