# The Drunkard’s Walk Explained

## 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.

# 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.

# 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.

# 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.

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