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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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