I’ve been playing a bit further with the visual demonstrations of the binomial/ central limit theorems. I love polymer clay as a medium, it’s extremely easy to work with, colourful, and you can even make fractal patterns with it.
Here’s my first approach, it turned out a bit less intuitive than I was hoping, but it was still fun to make.
The first step was to make a series of square stacks, encoding all possible 3-bit values (000, 001, 010…… 111):
Then slice them in quarters and make long strands from each:
Turn each one into a long rope:
And assemble into a Galton board style device, with each path represented by the colour ordering of the strand (e.g. 010 = Left, Right, Left)
You can immediately see that at the final clusters at the bottom, the number of each colour bands is the same (e.g. three blues, two blues and an orange, etc).
Representing that in the Galton board left-left-right leads to the same result as right-left-left.
I was fairly happy with that, but I wanted a more detailed model, and I decided to drop the encoded-strings idea as being to difficult to implement without distorting. Here’s my results for simple strings:
This I’m pretty happy with. You can see at a glance why the middle result is 6x more probable than the right hand result (since six paths lead there versus only one), and you also don’t have to wait for the balls to collect as with the conventional display.
After baking the models in the oven for half an hour they’re now robust and permanant, and I’ll keep them on the shelf for the next time the central limit theorem comes up.