The Drunkard’s Walk Explained

Stochastic Processes, Markov Chains & Random Walks

Brett Berry
Jun 12, 2017 · 7 min read

Stochastic Processes, Random Walks, & Markov Chains

This classic problem is a wonderful example of topics typically discussed in advanced statistics, but are simple enough for the novice to understand. The problem falls into the general category of Stochastic Processes, specifically a type of Random Walk called a Markov Chain.

Starter Calculations

Let’s get a feel for how these probabilities play out by crunching some numbers.

Possible probabilities after drunkards first 2 steps
Possible probabilities after drunk man’s first 3 steps

Deriving a Formula

This problem is only one of many variations. The probabilities 1/3 and 2/3 might as well have been any other probabilities summing to 1.

What is P2?

P2 is the probability of falling off the cliff on a path originating from 2 steps away. In order to fall off the cliff you have to move from 2 → 1 and from 1 → 0.

Formula describing the man’s chance of falling off cliff, P1

Solving for P1, the Probability of Eventual Doom

Now we have a quadratic to solve. All these p’s are a little confusing, so I’ll temporarily let P1=x to make the equation look more familiar to us.

Example of an invalid probability

What does this mean to us?

This means that we should model this problem with a piecewise function, where values for p less than 1/2 are modeled by x=1, values larger than 1/2 are modeled by (1 – p)/p, and p=1/2 can be modeled by either equation since they both yield x=1.

Back to our original scenario

Given a probability of 2/3 of stepping away from the cliff, and since 2/3 is greater than 1/2, we’ll plug it into the second solution to find the probability that the drunk man will fall off the cliff.

The Most Surprising Result

In fact, if his probability of stepping away from the cliff is less than or equal to 1/2, our function defaults to the P1=x=1 solution. Meaning that even at a 1/2 chance of stepping in either direction he is guaranteed to eventually fall off the cliff! There is no escaping it.


Math Hacks

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Brett Berry

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Check out my YouTube channel “Math Hacks” for hands-on math tutorials and lots of math love ♥️

Math Hacks

Tutorials with a fresh perspective.