Try This at Home:

Take a Random Math Walk

A random walk isn’t a quick way to move! But, it can be used to model and describe many other processes: from how a butterfly moves, to chains in polymer physics, and even stock market behavior!

What you’ll need:

  • A coin to flip
  • Paper and pencil
  • You!
  • Open space (like a yard or park)
  • A way to remember where you started (like a rock, sidewalk chalk, or other object)
walking in the grass

Here’s what to do:

  1. Grab your supplies and head outside.
  2. (Note: If it’s easier, you can do all the coin flipping sitting down and write down your results, then do the walking.) When you’re ready, put a marker down to indicate your starting point; a rock works great. Now, imagine you’re standing at zero on an invisible number line.
  3. Flip your coin. If you get heads, take one step straight forward on the number line (ex: from 0 to +1). If you get tails, take one step backward on the line (ex: from 0 to -1). Write down each of your coin flip results so you can trace your path. Example: If you get two heads in a row, you would take two steps forward in a straight line and end up at +2 on the number line. Try to make your steps about the same length apart, rather than deliberately making some steps farther apart or closer together.
  4. Continue flipping and walking at least 10 times. Try to make sure all of your steps are about the same length. Where do you end up?
  5. Try it again. Do you end up in the same place? Do you follow the same path?
  6. If you took 25 random walks, what’s the probability you’d end up back at zero?

Take it further:

  • After you take a few walks, can you make a graph, chart, or diagram of where you end up each time?
  • Try the random walk again but do it with friends or family, each person flipping their own coin and taking their own random walk. Do you all end up in the same place? How often do you pass each other?
  • Let’s add another dimension! Start flipping your coin again, but this time if you get heads, turn 90 degrees to the right and then step forward. If you get tails, turn 90 degrees to the left and then step forward. Try to use steps that are the same length each time, and make them big! Where do you end up after 10 flips? 25? How is this different from the 1-dimensional number line walk?

What’s going on?

A random walk is an interesting process in mathematics. When you flip your coin, you have equal probability (50/50 chance) of getting either heads or tails (moving forward or backward). Each time you take a random walk, the individual steps might be different, but if we put the results of many random walks together, we can calculate an average result (how far you end up from your starting point). The math is interesting, but a little complicated (check out the references), so we won’t explain it all here, but you end up using root-mean-squared distance. To simplify, if you take 25 steps, you end up, on average, 5 steps away from your starting point. The square root of 25 is 5! If you take 100 steps, how far away do you think you’d get? Well, the square root of 100 is 10, so only 10 steps away, on average. Read more about this concept in the references below!

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