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Question Number 5150 by FilupSmith last updated on 21/Apr/16
If we had a maze, which was contained  within the shape of a square, and it has  one enterence on its edge, and a centre  point from which you start, If you move  randomly, you will naturally escape.    Now, lets say the maze itself is generated  at completely random odds.    You start in the middle, and each move you  make (left, right, up, down), is random.    If I place another person in this maze  at a random point, and they too move  at random, which is the more likely outcome:  a) Person 2 lands on the same position        as you do  b) You escape the maze    (This question I thought of on my own  and am extremely curious on  everyones opiniom on this)
$$\mathrm{If}\:\mathrm{we}\:\mathrm{had}\:\mathrm{a}\:\mathrm{maze},\:\mathrm{which}\:\mathrm{was}\:\mathrm{contained} \\ $$$$\mathrm{within}\:\mathrm{the}\:\mathrm{shape}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},\:\mathrm{and}\:\mathrm{it}\:\mathrm{has} \\ $$$$\mathrm{one}\:\mathrm{enterence}\:\mathrm{on}\:\mathrm{its}\:\mathrm{edge},\:\mathrm{and}\:\mathrm{a}\:\mathrm{centre} \\ $$$$\mathrm{point}\:\mathrm{from}\:\mathrm{which}\:\mathrm{you}\:\mathrm{start},\:\mathrm{If}\:\mathrm{you}\:\mathrm{move} \\ $$$$\mathrm{randomly},\:\mathrm{you}\:\mathrm{will}\:\mathrm{naturally}\:\mathrm{escape}. \\ $$$$ \\ $$$$\mathrm{Now},\:\mathrm{lets}\:\mathrm{say}\:\mathrm{the}\:\mathrm{maze}\:\mathrm{itself}\:\mathrm{is}\:\mathrm{generated} \\ $$$$\mathrm{at}\:\mathrm{completely}\:\mathrm{random}\:\mathrm{odds}. \\ $$$$ \\ $$$$\mathrm{You}\:\mathrm{start}\:\mathrm{in}\:\mathrm{the}\:\mathrm{middle},\:\mathrm{and}\:\mathrm{each}\:\mathrm{move}\:\mathrm{you} \\ $$$$\mathrm{make}\:\left(\mathrm{left},\:\mathrm{right},\:\mathrm{up},\:\mathrm{down}\right),\:\mathrm{is}\:\mathrm{random}. \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{I}\:\mathrm{place}\:\mathrm{another}\:\mathrm{person}\:\mathrm{in}\:\mathrm{this}\:\mathrm{maze} \\ $$$$\mathrm{at}\:\mathrm{a}\:\mathrm{random}\:\mathrm{point},\:\mathrm{and}\:\mathrm{they}\:\mathrm{too}\:\mathrm{move} \\ $$$$\mathrm{at}\:\mathrm{random},\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\:\mathrm{more}\:\mathrm{likely}\:\mathrm{outcome}: \\ $$$$\left.{a}\right)\:\mathrm{Person}\:\mathrm{2}\:\mathrm{lands}\:\mathrm{on}\:\mathrm{the}\:\mathrm{same}\:\mathrm{position} \\ $$$$\:\:\:\:\:\:\mathrm{as}\:\mathrm{you}\:\mathrm{do} \\ $$$$\left.{b}\right)\:\mathrm{You}\:\mathrm{escape}\:\mathrm{the}\:\mathrm{maze} \\ $$$$ \\ $$$$\left({This}\:{question}\:{I}\:{thought}\:{of}\:{on}\:{my}\:{own}\right. \\ $$$${and}\:{am}\:{extremely}\:{curious}\:{on} \\ $$$$\left.{everyones}\:{opiniom}\:{on}\:{this}\right) \\ $$
Commented by prakash jain last updated on 23/Apr/16
Let us say you have n×n grid.  To create a random maze you choose some  of the n^2  line and put walls on those lines.  This means a number m is randomly chosen  between 1 to n^2 . then m walls are created  by randomly choosing m lines in n^2  grid.  Calculating probability of the person  moving randomly will need be calculated  for each case m and then a conditional probability  will need to be summed.  more work to be done.
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{you}\:\mathrm{have}\:{n}×{n}\:\mathrm{grid}. \\ $$$$\mathrm{To}\:\mathrm{create}\:\mathrm{a}\:\mathrm{random}\:\mathrm{maze}\:\mathrm{you}\:\mathrm{choose}\:\mathrm{some} \\ $$$$\mathrm{of}\:\mathrm{the}\:{n}^{\mathrm{2}} \:\mathrm{line}\:\mathrm{and}\:\mathrm{put}\:\mathrm{walls}\:\mathrm{on}\:\mathrm{those}\:\mathrm{lines}. \\ $$$$\mathrm{This}\:\mathrm{means}\:\mathrm{a}\:\mathrm{number}\:{m}\:\mathrm{is}\:\mathrm{randomly}\:\mathrm{chosen} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{to}\:{n}^{\mathrm{2}} .\:{then}\:{m}\:\mathrm{walls}\:\mathrm{are}\:\mathrm{created} \\ $$$$\mathrm{by}\:\mathrm{randomly}\:\mathrm{choosing}\:{m}\:\mathrm{lines}\:\mathrm{in}\:{n}^{\mathrm{2}} \:\mathrm{grid}. \\ $$$$\mathrm{Calculating}\:\mathrm{probability}\:\mathrm{of}\:\mathrm{the}\:\mathrm{person} \\ $$$$\mathrm{moving}\:\mathrm{randomly}\:\mathrm{will}\:\mathrm{need}\:\mathrm{be}\:\mathrm{calculated} \\ $$$$\mathrm{for}\:\mathrm{each}\:\mathrm{case}\:{m}\:\mathrm{and}\:\mathrm{then}\:\mathrm{a}\:\mathrm{conditional}\:\mathrm{probability} \\ $$$$\mathrm{will}\:\mathrm{need}\:\mathrm{to}\:\mathrm{be}\:\mathrm{summed}. \\ $$$$\mathrm{more}\:\mathrm{work}\:\mathrm{to}\:\mathrm{be}\:\mathrm{done}. \\ $$
Commented by FilupSmith last updated on 24/Apr/16
That is very similar to my usumption.  Thanks!
$$\mathrm{That}\:\mathrm{is}\:\mathrm{very}\:\mathrm{similar}\:\mathrm{to}\:\mathrm{my}\:\mathrm{usumption}. \\ $$$$\mathrm{Thanks}! \\ $$

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