Feynman’s Random Walk: Understanding Movement by Chance

In Chapter 6-3 of The Feynman Lectures on Physics, Richard Feynman introduces us to a simple but powerful concept: the random walk. It’s a playful idea on the surface — a game involving coin tosses and small steps — but beneath it lies a deep insight into the nature of chance, measurement, and even atomic motion.

A Game of Steps

Imagine a player starting at a central point. At each turn, they flip a coin: heads means a step forward, tails a step back. With each toss, their position changes. It’s entirely random, so after many steps, where do we expect them to be?

Feynman’s first answer is wonderfully intuitive: probably not far from where they started, but not exactly at the centre either. Sometimes they might wander off quite a bit, other times not much at all. If we were to track many such walkers, the average position would still be zero, because they’re just as likely to move left as right. But the average distance from the starting point, regardless of direction, grows with each step.

He shows this through a graph that tracks three random paths (taken from real coin tosses) — they zigzag unpredictably, but each tends to drift away from the centre over time.

Measuring the Wander

Rather than trying to calculate the exact location of a walker after many steps (which is mostly meaningless in a random system), Feynman focuses on how far they’ve wandered on average. To do this, he looks not at the direction, but the overall spread — a measure of how “lost” a walker might typically get.

As the number of steps increases, so does this average wandering. It’s not a straight line — the drift grows more slowly than the number of steps — but the key point is this: the more steps you take, the more likely you are to be further from where you started.

Feynman likens this to Brownian motion, the jittery movement of tiny particles in a fluid, and even to the way errors stack up in repeated measurements. In both cases, random variation builds over time — not in a predictable direction, but in overall size.

Coin Tosses and Patterns

Feynman connects this to a coin-tossing game he discussed earlier. Every coin toss is a step: heads moves you forward, tails back. If you toss the coin a large number of times, the results will mostly cluster around the middle — that is, an even split of heads and tails. But there will always be some spread, a natural amount of “wiggle room” in the results.

He shows that this variation is not random chaos — it has a predictable pattern. The more tosses you make, the narrower the relative variation becomes. The graph of outcomes forms a bell-shaped curve, with most results falling near the middle and fewer as you move outwards.

In one example, he looks at 30 steps and shows that most walkers end up within a few steps of the centre. This aligns beautifully with the curve’s shape and confirms the underlying principle: even randomness has structure.

Can You Trust a Coin?

This leads to an interesting practical question: how do you tell if a coin is fair?

According to Feynman, if you toss a fair coin many times, the number of heads and tails should roughly balance out. But “roughly” is the key word — it won’t be exact every time. Sometimes you’ll get streaks of heads or tails just by chance.

So how far off from a perfect split is “normal”? Feynman gives a way to estimate this natural wiggle. If your results deviate only slightly from what you’d expect, there’s no reason to be suspicious. But if the coin consistently lands one way far more often than the other, it might be biased — or perhaps the person tossing it has some clever trick up their sleeve!

Still, he cautions: probability doesn’t give proof. Even a fair coin can behave oddly now and then. Statistics only tell us what’s likely, not what’s certain.

The Nature of Probability

Feynman finishes with a powerful reflection on what probability really means. It isn’t some fixed truth hiding behind the numbers — it’s a way of describing our uncertainty. What we call “chance” reflects our incomplete knowledge. The more we observe, the better our estimates become, but there’s always room for surprise.

In science, especially in experimental physics, we deal with probabilities all the time. But it’s important to remember they are never absolute. They evolve with new data and deeper understanding.

Final Thoughts

Chapter 6-3 is classic Feynman: he takes a childlike game and turns it into a lesson on the deep structure of randomness. Whether it’s atoms bouncing in a gas, errors adding up in measurements, or the fairness of a coin toss, the random walk provides a beautifully simple framework for understanding it all.

It’s a stroll through the unknown — guided not by certainty, but by pattern, probability, and Feynman’s signature curiosity.