DIR Return Create A Forum - Home
---------------------------------------------------------
gworld
HTML https://gworld.createaforum.com
---------------------------------------------------------
*****************************************************
DIR Return to: MATH FACTS
*****************************************************
#Post#: 117--------------------------------------------------
DRUNKEN WALKER AND FLY
By: eba95 Date: July 30, 2010, 6:50 am
---------------------------------------------------------
Drunken Walker and Fly
Imagine a drunken person
wandering on the number line who
starts at 0, and then moves left or
right (+/-1) with probability 1/2.
What is the probability that the
walker will eventually return to her
starting point Answer: probability
1.
What about a random walk in the
plane, moving on the integer lattice
points, with probability 1/4 in each
of the coordinate directions? What's
the chance of return to the starting
point? Answer: also probability 1.
OK, now what about a drunken fly,
with 6 directions to move,
probability 1/6? Surprisingly, it is
probable that the fly will never
return to its start. In fact it only has
probability around 1/3 of ever
returning.
Presentation Suggestions:
Try to give a little insight by
illustrating a random walk on the
line for several steps.
The Math Behind the Fact:
A probabilist would say that simple
random walks on the line and plane
are recurrent, meaning that with
probability 1 the walker would
return to his starting point, and that
simple random walks in dimensions
3 and higher are transient, meaning
there is a positive probability that he
will never return! This is because
there is so much "space" in
dimensions 3 and higher.
*****************************************************