Walk the grid
You start at the point in a Cartesian coordinate system.
Every second you move randomly 1 unit either up, down, left or right.
What is the expected value for the number of seconds it will take for you to land on a point where and are integers ?
Clarification : You can walk anywhere in all four quadrants of the Cartesian coordinate system. and/or can be negative.
Image credit: clipartfreefor.com
The answer is 30.
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1 solution
By symmetry there are only five unique expectation values in the problem.
They are represented by the expectation values from the following points:
( 1 , 0 ) , ( 1 , 1 ) , ( 2 , 0 ) , ( 2 , 1 ) , ( 2 , 2 )
Let E 1 through E 5 be defined as the expected value for the number of moves it would take to get to ( 5 m , 5 n ) for integral m and n from each of these locations respectively.
From each grid point the expectation value is given by E = 1 + 1 / 4 ∗ (sum of neighboring expectation values)
So, we have the following set of linear equations:
E 1 = 1 + 0 . 2 5 ( 2 E 2 + E 3 ) E 2 = 1 + 0 . 2 5 ( 2 E 4 + 2 E 1 ) E 3 = 1 + 0 . 2 5 ( E 1 + E 3 + 2 E 4 ) E 4 = 1 + 0 . 2 5 ( E 2 + E 3 + E 4 + E 5 ) E 5 = 1 + 0 . 2 5 ( 2 E 4 + 2 E 5 )
Sovling for E 2 , this gives, E 2 = 1 8