Law of Large Numbers Simulation Exploration Activity

The concept behind the Law of Large Numbers is one that our brains struggle to come to terms with. It just doesn’t naturally sit in our heads. I have seen many curricular materials that introduce this Law with applet simulators that do some repetitive chance task, like rolling dice or flipping a coin, hundreds and hundreds of times in the blink of an eye. They are GREAT to illustrate the point, BUT.

Just the other day I had a student talking to me about his DnD game and how he needed to get some D20’s and other non-standard dice so that he could stop using virtual dice because they were, and I quote, “less random”. Of course we then had the discussion that the virtual dice are actually programmed to be just as random as you rolling a physical die, but the point is that our students’ brains do not think that! So, when you use the applet, they’re like “cool illustration, bro, but my brain knows what happens when you REALLY flip a coin.” So much of the Probability and Statistics curriculum is this little fight against our brains.

So I decided to come up with a physical way to illustrate the Law of Large Numbers, in the hopes that my students’ brains would be more accepting of the results if they had physically interacted with the probabilistic material. I’ve used coin flipping labs in the past in Algebra 1 to model exponential decay, and I figured I could adapt just a few parts of it to make it work for illustrating the Law of Large Numbers instead.

We started with 4 basic questions that my students pretty easily answered – I gave them a few minutes to think on these and then we discussed them as a class.

Then, I handed each of them a Dixie cup with 10 pennies in it. This isn’t that big of an investment (even if you have class sizes bigger than mine, it might cost you $3.00 or so to get this many pennies?), but you could also use counter chips or something that is different on both sides that you have at hand. I asked them to make sure their cups had 10 coins, since I’m not always the greatest at counting, and then they were instructed to go through stages of flipping different numbers of coins. We did stage 1 together, so they would know how to fill out each column. With our COVID year, I’m out of practice at capturing pictures of students while they work, so I just have their worksheet responses…

When they got to stage 6, I had created a simple Google Sheet for them to put their data in, and programmed it to automatically calculate the totals for # of heads and coins flipped.

Then, we graphed each stage of our results. In the pictures, you’ll see that when I originally created this, I did not calculate the total coins flipped for our whole class combined correctly, so the x-axis does not go high enough to graph all of our stages. I graphed this on my SMART board projected onto my white board with my own data, so I went off the edge of the screen to show my students what that last point would look like, but I also adjusted the worksheet to fit my typical class sizes for the future – if you need a longer x-axis for yours, I just took a screenshot of a Desmos graph I created with the scales I wanted!

Question 7 has them draw a horizontal line at 50% heads. After we did this, we discussed how our graphs looked slightly different but what they all had in common – as one student put it, they “started hugging the 50% line as it went further to the right” which I loved!

We concluded by introducing the vocabulary of experimental and theoretical probability and summarizing what the Law of Large Numbers says.

I think this activity did really help my students internalize the LLN and combat their brain’s incorrect thoughts a bit – I got better results this term on our assessment questions regarding experimental vs. theoretical probability and the LLN than I have before!

In the future, I might combine this with the use of one of the simulation applets – once we’ve physically done it with actual coins, show them what does happen when you keep flipping thousands more times. I think that might solidify the ideas in their minds even more.

Here is the resource I created for students – it will force a copy of your own so you can make any changes you want! I’ll also link a forced copy of our class Google Sheet so you can see how that’s set up and it should work for any number of students!