Random Walks

Public summary: 

See how biased coins and random chance can lead to surprising conclusions.

Choosing direction using a coin flip.
Useful information
Kit List: 

* Duplo blocks
* Dry wipe pen
* Duplo baseplates (2 stuck together for the race)
* Fair coin (10p)
* Biased coin (2p)
* Selection of Dice (Biased and unbiased)
* Print out of a monopoly board

(n.b. if there isn't a biased coin, because it's got lost for any reason, you can always just use a bit of sleight of hand and lie, saying it's heads two thirds of the time...)

Packing Away: 

Place into the maths box, bag the dice and coin sensibly so they're easy to find.

Frequency of use: 

A variety of experiments can be done with random walks, showing not only the idea of a biased probability, but also that even when an outcome is weighted one way, random chance can lead it to come out the opposite.

There are a few fun games to play here. Most involve building up individual children's results into a nice smooth distribution, by building duplo bricks with their names on them into a nice smooth wall.

Explanation and Demonstrating

The simplest thing to start with is with a brick on the middle of a baseplate and an unbiased coin. If you set the rule that when the brick gets to an edge it falls off and the game is over, that a head moves you in one direction, and a tails in the other. They should see that there's a good bit of back and forth, but eventually you should end up dropping off one side or the other.

Then try doing the same with the biased coin (roughly 66% of the time it gives heads) making sure the coin is allowed to land on the table (preferably on a piece of paper to damp bounces and speed up the whole process), show that if repeated a few times the brick has a tendency to fall off the same side.

Alternatively, as the first method can sometimes take too long, use about six bricks build a tower: placing the next brick on top of the previous one and one set of teeth to the side depending on the toss being heads or tails. Then repeat this a couple of times with both coins and you should see that with the biased coin the tower of bricks tends to lean to one side.

Now set up a few bricks, all starting different distances along the board, and watch which one is the last one left, it should be the one initially roughly two thirds of the way along, not as many people might assume when first introduced to a biased coin, as far from the side it's biased towards as possible.

If you can't find the coin then there are also some dice, use the convention even moves one way and odd the other. There are several non-standard dice which can be used for a biased coin.


Finally, you can build a really lovely distribution by starting the block at one end of two mats put together and using the biased coin. With every flip you move the block one space forwards (one block length, not one set of teeth) and one space in whichever direction the coin dictates. When the blocks reach either side, that's where they stop. Keep everyone else's blocks on the board as well, and a nice distribution should build up. It should look, very roughly speaking, like a pear cut in half, with a bulge at the near end and then at tail trailing off. Write their name on the block with the dry wipe pen, you'd be surprised how keen they are to come back later and see how the distribution's evolved around their block.

You should see roughly the same distribution on both sides, just with significantly less blocks on the side that is biased against. You can discuss with them things like why there are no blocks at all in the first few spaces, why there is a peak (it roughly corresponds to the balance between a point which there are a lot of various paths to, and where the probability the brick hasn't already fallen off the board, balance out).

--ALTERNATIVE PLAN-- (experiment still developing so let us know what works best!)

Some demonstrators, rather than building up a distribution over the day, which can take a long time and still won't look exactly smooth even after all the pieces have been used, prefer to just trace the path of one random walk, placing a new brick for each step. This gives a clearer memory of the rambling nature of the walk, and there should be sufficient bricks for two or three paths simultaneously.

This also helps defend against the odd child who comes along and tries to tear the whole thing down...

It may be the better options for events with younger children, where the concept of a distribution is a bit much, and also it just looks really cool!

--Real World Applications--

If you fancy it, theirs a great analogy between this experiment and genetic drift. Genes that develop in populations of animals normally have some selective bias for either adoption throughout the population or deletion. But genetic drift is the process of random genetic evolution, and has a strong analogy in a biased coin toss. From this it's possible to see that even a positive evolution (one that natural selection favours some percentage of the time) may be deleted by chance, or a deleterious one spread throughout the entire population.

The key idea here is that when there is one or more boundary, i.e. the edge of the board, we see markedly different behaviour to that we'd expect for an infinite number of tests, where deleterious mutations would never remain in the population and we'd always expect to the brick to end up on the side that it's biased towards eventually.

There are many other examples of random walks in nature, like photons in the solar atmosphere which take thousands of years to complete a 'drunkards walk' from the centre to surface of the sun and only eight minutes to make it all the way out to earth. More complex examples include the game of life, a simulation of how populations grow, receded and migrate over time. You could even mention the fact that for some random walks, even in infinite time you would not expect it to pass through all points.

If you waned to link to something else you can use the monopoly print out to demonstrate a random walk on the board. If you land on community chest or chance there's another random element you could move else where but it's possibly best to ignore this. If you roll 3 doubles in a row you go to jail. If you can find put properties of this walk you'd be able to gain an advantage at this game. You can see the Kruskal's Count experiment for some other ideas related to this.

By flipping two coins you can do a random walk in 2d, ask people if you think you'd be able to get back to the start, will it take longer to fall off an edge. Then repeat in 3d where this isn't true.

Risk Assessment
Date risk assesment last checked: 
Wed, 05/02/2020
Risk assesment checked by: 
Conor Cafolla
Date risk assesment double checked: 
Wed, 05/02/2020
Risk assesment double-checked by: 
Risk Assessment: 

See how biased coins and random chance can lead to surprising conclusions.

Hazard Risk Affected Person(s) Likelihood Severity Overall Mitigation Likelihood Severity Overall
Duplo blocks/coins/dice Swallowing Duplo blocks and coins. Small children 2 5 10 Duplo blocks should be sufficiently large that they are safe from swallowing for all ages, but keep an eye out just in case. Don't let small children handle coins. Keep an eye on all the equiptment.
Call first aider if child swallows, if choking encourage child to cough.
1 5 5
Losing objects Tossed coins / thrown dice flying away and can either hit someone, or get lost e.g. under tables, resulting in kids running after them and hitting something. All 3 3 9 Check you can flip a coin without losing control of it, otherwise just spin it on the table. It's perfectly fine to let the children toss the coin themselves, but make sure they're capable of doing it safely with a trial flip first. Don't let children retrieve lost coins
Call a first aider in the event of injury.
2 3 6
Objects on the floor Slip hazard if things dropped on floor. All 3 3 9 Keep an eye on where any objects go, and try to confine them to a desk or fixed area. Pick up anything that falls immediately. Do not let multiple unattended children use objects at the same time.
Call first aider in case of injury.
1 3 3
Pens Children may be tempted to lick the marker pens and fall ill as a result. Children 3 3 9 Keep an eye on the pens and don't let small children use them without supervision. In the event of an incident, tell parent to contact a GP if child feels unwell later. 1 3 3
Experiment photos: