Random Walk, explained.
A random walk accumulates successive random steps, so an unbiased step mechanism can produce paths that end far from the starting point.
Why it happens
Equal chances of moving left and right give zero expected position, but squared distance increases with the number of independent steps. Positive and negative endpoints can cancel in a mean even when each traveler is far from home.
A simple random walk adds independent random steps. A zero expected step does not force a particular path to stay close to its starting point.
Read the result
Resample and compare mean position with mean absolute distance. The chart displays five paths, while statistics use twenty. The square-root reference is theoretical root mean squared distance, a different summary from mean absolute distance.
A worked example
Balanced steps without a balanced outcome
A walker makes 100 independent steps, each one unit left or right with equal probability.
Expected final position is zero, while root mean squared position is ten units.
The neutral average does not promise that a particular traveler returns home or remains near the origin.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Twenty one-dimensional paths start at zero and independently move +1 or −1 with equal probability. The highlighted path is one sample, and all paths remain unbounded. The root mean squared position reference is sqrt(number of steps); sample averages can fluctuate.
Where this idea is useful
A practical use
Repeated small, balanced changes can still accumulate into substantial variation. Distinguish an unbiased mechanism from a guarantee of a neutral realized outcome.
A common misconception
“A fair process must soon compensate for its past drift.”
Independent steps have no corrective memory. An accumulated imbalance does not change the probability of the next step.
What this explanation leaves out
- No drift or dependence is modeled. Finite runs do not prove eventual return; Pólya's recurrence results concern particular infinite-horizon random walks.
Does this experiment prove eventual return?
No. Finite paths cannot establish an infinite-horizon recurrence result. The theorem's conditions, including dimension and step rules, must be specified separately.
Does your average hide large distances in opposite directions?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.