Random WALK

A man gets out of a bar. He is too drunk, and doesn't know where he is going to.

At each step he chooses a random direction, North or South, with equal probabilities.

What does his path look like? How far from the bar is he after 1000 steps? N steps?

Let's watch him in quick and slow-motion. The picture shows his browsing unwrapped

in time, like snap shots. Click to see his random walk animated :

Watch carefully in "Slow-Mo". Do you see any pattern?

In "Fast-1" mode, do you see any pattern for the final location?

How far is the drunk man from the bar after some number of steps?

At each moment, can you predict the next step (up or down)? No!

Can you predict if his final location is at North or South? No!

But we can make some statistical predictions!





The average location of a drunk man is at the bar.

Average Distance from the bar is SquareRoot of the Number of Steps.

Chances to be within "3-Sigma spread" from the bar is about 99.7%.

Your computer/browser must be FAST enough to show 10,000 walks.

Exactly the same picture we get in simple coin tossing game (head or tail).

Each lightning shows the amount won/lost through a game (of 1000 tossings).

Coin tossing game is a possible class - ACTIVITY.

Any conclusion? If each bet is a cent per tossing, then in a game

of 1000 tossing it is almost impossible to win/lose over a buck.

Random Walk relates directly to the Pascal's Triangle. HOW?

Remember Pascal's Triangle? It's in the right... Start at the top cell. From each cell you can go down left or down right, due to gravity. There are two steps to do down, and from each new cell there are also two steps down... These ways branch like a "binary" tree. It's also the well known game on the board of fortune, with islets, nails, and rolling balls...

But, how many ways are there that take us from the top cell to a chosen cell below? Obviously, they are given by the Pascal's number of that final cell. WHY?

So, the chances to end up in a cell at some lower row far are given by the cell number relative to the total number of ways leading to the bottom row. And this trickling down avalanch demonstrates exactly the above Random Walk process. The binomial coefficients describe so called Binomial Distribution, which turns into Gaussian, or Normal Distribution for large number of rows.

It is amazing how different physical processes relate to each other - through MATH!

Send questions and comments to mamikon@caltech.edu