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Random Walk Applet
1 dimension: The blue point is moving on a line with integer coordinates: The 2 directions of a single step: x+1, x1 2 dimensions: The blue point is moving in a plane with integer coordinates: The 4 directions of a single step: x+1, x1, y+1, y1 
select from the menu 

button starting a single walk, maximum of n=1000 steps, the diagram at the bottom is showing the distances d(n) button to stop the walk 
1 dimension:
An interesting question arising in the
study of random walks concerns
the probability of returning to the initial position
(origin, "equalization").
The probability P(n) of return to origin at step n (n even) is:
For large n (even):
Graph of the first (strict) formula:

Applet results:
The total number of
returns to origin (within a fixed number n of steps) is
proportional
to the number N of walks:
The probalibity for n=100 steps is 0.076
2
dimensions:
Example:
button starting a set of N walks 

the numbers of steps and walks can be selected from the menus 
Books 
Küppers, BerndOlaf: Die Berechenbarkeit
der Welt, Grenzfragen der exakten Wissenschaften. S.
Hirzel, Stuttgart 2012. Entropie und Zeitstruktur, S. 200210 Eigen, Manfred, and Winkler, Ruth: Das Spiel, Naturgesetze steuern den Zufall. Pieper, München 1975. Kapitel 4:Statistische Kugelspiele 

Random
Walk1Dimensional (Wolfram MathWorld) Random Walk2Dimensional (Wolfram MathWorld) A 1D Random Walk Visits The Origin Infinitely Often 
Updated: 2012 Sep 22