Talking about Pólya’s Urn with kids – inspired by Jim Propp’s blog post

A few weeks ago I saw another great blog post by Jim Propp – this one was about Pólya’s Urn:

Jim Propp on Pólya’s Urn

I’d meant to talk through it with the boys, but it was one of the topics that slipped through the cracks.

However, I was reminded of Propp’s post yesterday when Steven Strogatz shared the list of new fellows of the American Mathematical Society and Propp’s name was on the list:

So, this morning we talked about Pólya’s Urn. This is a little more risky Family Math talk than usual since I didn’t go back through Propp’s post before having the conversation with the boys. My intention was to highlight some basic ideas just to see what the kids had to say. I think we had a great conversation, and I sure hope that I got the main details right! (and, in all seriousness, if you want to talk about Pólya’s Urn with anyone, study Propp’s post, not this one.)

We started off by talking about some simple ideas about the binomial distribution (without naming it). The idea here was to fill an urn with either black or yellow balls by flipping a coin at each step. The boys had really interesting ideas about how you could fill an urn this way.

Instead of flipping a coin, we decided to use a 6-sided die since (as I learned the hard way) flipping a coin on a video doesn’t work so well.


After the short talk (and one example) about the binomial distribution in the last video, we spent the next part of the talk exploring the distribution in a bit more detail. The boys were surprised to find Pascal’s Triangle make an appearance in the discussion here.

For me the most interesting piece of this part of the project was listening to the kids try to find the right language to describe probability. There are a lot of moving parts even in this relatively simple example, and just finding the words to describe what’s going on stretches them a bit.

But they get there, which was really nice 🙂


With the coin flip example out of the way, we moved on to talking about Pólya’s Urn. The first part of this talk explains the new process and why the process begins with the Urn already having a single black and two yellow pieces.

One interesting piece of math that comes up in this part of the project is this – how can we do a random simulation of this process? The kids find a couple of different ways to use the various dice we have handy in the simulation. I was happy to see a few different ideas from them here.

After figuring out how to use our dice, we ran one simulation and discussed what we saw (which included the phrase “unless a miracle happens”- ha ha).


Next, we tried to replicate the process for the binomial distribution that led to the surprise appearance of Pascal’s triangle. The process is a bit more complicated here, but the boys were able to find a great way to describe it. Working through the binomial distribution earlier in the project was a great help here.


At the end of the last video we’d found the first 3 rows describing the process we followed filling up Pólya’s Urn. They boys were starting to wonder what the next row would look like, so I thought working out the next row would be productive.

Once we wrote down the next row I asked them what patterns they saw. Just as in Pascal’s triangle, there are lots of interesting patterns.


So, a fun project about an idea from probability theory that I’d not seen prior to Propp’s post (with super embarrassing apologies to him if he discussed it when I took his combinatorics class in 1992 – ha ha!!) . It was exciting to be able to share this idea with kids and it seems like a really fun way to get younger kids thinking about basic ideas in counting and probability.

Also, I remember this post from Patrick Honner after he read Propp’s piece:

Funny, indeed, how that works 🙂


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