The Mathematics of the Drunken Walk

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Lectures on statistics are not always the most exciting ones in the world of maths however certain subjects tend to attract students attention.  One of those is of course alcohol, and not in the over indulgence and problematic way where you end up taking a drug like this one called Selincro to combat it’s effects.   It’s actually revolved around a concept called the Drunkard’s Walk a famous mathematical concept.

It can be best explained in a theoretical example about a drunken man who was walking way too close to a cliff for someone in that state.  The idea is that from his starting position a single step forward would send him over the cliff.   He takes completely random steps oblivious to his own safety in any direction.  His probability of taking a step away is 2/3 and of taking a step towards the cliff is 1/3.  The mathematicians problem – what are the drunkards chance of escaping the cliff?

It’s a classic problem but actually touches on some advanced statistical topics. The particular topic is centered around Stochastic Processes which covers these ‘random walk’ issues, the specific name is called a Markov Chain.

Stochastic Process – a random process which explains how a system or process changes over another unit (commonly time).

Random Walk – a path derived from a series of completely random steps in some defined mathematical space.  Our example is the very drunk man tottering on the edge of a cliff.

Markov Chain – a random walk which actually maintains independent events.  That is the next event is not dependent or related to the previous one.  The drunken man has to be so drunk that his position and last action has no bearing on his next step!

The mathematics of this situation is of course all related to probabilities and how likely the man is to survive his reckless behaviour.   The simplest point is the beginning where he is one step away from the edge, the probability of surviving the next step is 2/3 and he has a 1/3 chance of stepping over the edge.

After that of course it get’s more difficult as the man if he survives will be moving away from the cliff edge and buys himself some time.   The easiest way to visualize this situation is to draw a chart of the probabilities with all the possibilities.  This has to include his relative position from the cliff and an assumption about where he ends up and what position is safe!

The problem is actually not that complex but it can seem so purely because there are so many possibilities after the initial even.  The secret is to define the chart with the possibilities and then try and generalize the problem in order to create a formula. This has to include the probability of stepping towards the cliff edge and stepping away.

To solve these problems you normally define the expected probability of the event you are trying to measure.  So in this case it would be defining the probability of falling from the cliff – say P1.

Without too much detailed analysis we can get to the formula as follows:

P1 = (1-P) + (p*P2)

Here the variable P2 is the probability of falling from the cliff on a path consisting of 2 steps!

John Welcome