Power and Virtual Coins

This activity was inspired by the article “Innocent Until Proven Guilty”, by Catherine Case and Doug Whitaker. NCTM Mathematics Teacher, Volume 109, Issue 9 (May 2016)

Around February each year, the AP Statistics message boards come alive with new and veteran AP Statistics teachers seeking ideas to help students understand the concept of statistical power. While Power is a “minor league” topic in the AP Stats curriculum, a robust discussion of the concept can help tie together the logic of statistical inference: P-values, error and sampling variability. I’ve developed a few activities to try to bring Power to life (see here and here). And while each was satisfying in their own way, none of them really met one of my overarching classroom goals – to have students identify and express a new idea with their groups before I provide clarification. This year’s activity worked nicely as it allowed students to experience statistical power and generate meaningful conversation. Download the student version below, then read to learn how it works.

In this activity students will investigate the “fairness” of 3 virtual coins through a Desmos graph, using 3 different sample sizes to compile evidence. For each sample, students use their graphing calculator to compute a P-value and then reach a statistical conclusion. For coin A, I led students through the steps for n=10 and encouraged them to work through the next two sample sizes using their group-mates as a support system.

As students completed all three columns for coin A, I asked them to make a final decision regarding the fairness of coin A – is there convincing evidence that coin A is unfair? Students discussed findings with their groups and thoughts about how each column provided convincing evidence. Here is what the class-wide vote and conversation revealed:

  • Of my 42 total students (2 classes), only 1 student concluded that coin A was unfair.
  • All groups agreed that the larger sample size (n=100) was more useful in reaching a decision about the coin.

Spoiler alert: coin A is unfair! If you take a peek under the Desmos hood, you will find that coin A is “programmed” as 48% heads, 52% tails. I didn’t reveal the true proportion until the end, but we are off to a good start here: small differences between the null and “truth” are less likely to be detected.

Groups then tackled coin B with little assistance from me. Working through each column, then the follow-up conversation and decision, took about 5 minutes. This time about 60% of the students concluded that coin B was unfair.

Finally, coin C. Many students quickly concluded that coin C was unfair (it is!) but worked through each of the columns and sample sizes. In the end, there was class-wide agreement that coin C is an unfair coin.

At this point I revealed the truth about each coin:

  • Coin A: 45% heads
  • Coin B: 40% heads
  • Coin C: 25% heads

So, what do our finding show us about hypothesis testing and decision-making as a whole? I was thrilled when one of my students who does not volunteer often raised his hand to offer the following: “If there is a big difference between the null and the truth, it’s easier to reject the null.”

Yes! That’s a big part of power. What else?

Larger sample sizes are more likely to detect a difference when one exists.

Yes! And now we have a nice framework for power. From here I shared a working definition of power and included thoughts on alpha, which are not part of this activity now but could be in a later version.

EmPower your students to develop statistical ideas!


Chessboards for Jack

“Mr. Lochel – we have a check to send to you. Please provide us an address.”

What a fantastic e-mail I received this past year. It combines two wonderful things – money, and people wanting to send money to me. But this money was coming from a very special place.

In 2017, a talented, driven group of students at my high school participated in the Moody’s Mega-Math Challenge. In this annual math modeling contest students are provided a prompt and 14 hours to develop a framework for a solution. The contest is now sponsored by MathWorks, and you can learn about the contest and resources for getting your students involved at the contest website. The team earned an honorable mention award for their solution – and a $1,000 prize to be split amongst the team members. Not a bad way to close out their math careers in our district, and on to bigger and better things!

The 2017 Math Challenge Team: Principal Dennis Williams with Quinn, Jack, Mathew, Jason and Dmity.

Jack enrolled at Dartmouth University in the fall of 2017. Like most students who move on to college, I had limited contact with Jack after he left high school. And similar to many of my colleagues here who were part of Jack’s life, I was gutted when I learned of his passing in 2019.

Fast-forward to last spring, when the nice folks at the Society for Industrial and Applied Mathematics informed me that 2 years after winning his prize for the modeling contest, Jack hadn’t cashed the prize check. After contacting his family, the funds were donated to the high school math club. A wonderful gesture, but math club really doesn’t require much funding….so what to do with found money? How best to honor Jack’s legacy at our school?


Red and Black week chess tournament. Jack is seated at the center table, in black.

I’m not sure what percentage of Jack’s down time at our high school was spent playing chess, or even how much time was spent playing chess when he was supposed to be working on something else, but it was a solid amount. But if you observed Jack playing chess you quickly realized the game was secondary. Having a conversation, thinking deeply, learning about someone new – these were the important residuals of a chess board.

I used the donated money to purchase as many nice-ish chess boards as possible and worked with our HS guidance department to generate a list of teachers Jack worked with during his time in our district. This past week I organized 8 boards including a note about Jack encouraging players to make a new friend and enjoy a conversation through chess. Four of Jack’s former teachers received chessboard gifts, and classrooms all the way down to 4th grade will have a new set to enjoy. I hope the wonderful culture of caring about others and getting to know someone new through chess which Jack embraced will live on through these boards.

Getting the boards ready for delivery.

I hope you find these resources helpful for learning more about Jack and helping individuals and families in your own community who struggle with mental illness.

Jack Duffy 5K Run

NAMI – Montgomery County PA

Liv Associates

Thanks Jack.

Statistics Technology

Adapting to Remote Learning – Egg Roulette

This is the first in what will likely be a series of posts about classroom moves which I have adapted for remote learning. I hope you enjoy them!

In my freshman-year Prob/Stat course, students experience a probability lesson featuring the game “Egg Roulette”, based on a bit from Jimmy Fallon’s Tonight Show. Here is a summary of the “live” lesson: This year, there were two considerations for how I would have students investigate the game: conducting the simulation and collecting the results.


The first class simulation involves two unsuspecting volunteers and my actual container of 12 “eggs” – filled with little fuzzballs. Click the link in the last paragraph to see a video of how it works. In the main simulation, students use decks of cards to play the game repeatedly. Give pairs of students 13 cards all of the same suit. Discard the ace. Then, the 10, jack, queen and king represent “raw” eggs. The other cards represent the “hard-boiled” eggs. In a remote environment I could have used a site like to draw cards, but I also saw an opportunity to build a simulation students could use to quickly analyze repeatedly. This Desmos link allows students to play the many times: Click the “rerandomize” button to generate repeated plays of the game. Online, we talked through a few of the simulations and I found the students quickly understood the format.

Jimmy Fallon Egg Roulette simulation on Desmos


I have used a number of methods for collecting class results over the year: sticky dots on a poster, Post-It Notes on a wall, digital data collection. Clearly this year we had to go digital, and the site came to the rescue. New this year, teachers have a “collaborative” option – this feature generates a class code from which students can submit their data to the class (thanks Josh Tabor and Luke Wilcox!). The results update in real time. Each student then pasted the class graph into OneNote and a discussion of Jimmy Fallon’s “meanness” – is he being nice to his guests by letting them draw first? – followed.

Results of our class simulation using stapplet

The rest of the lesson and discussion felt similar to previous years. I challenge small groups to find the probability of a player losing in round 3. This leads us to probability ideas of independence / dependence and the multiplication rule. The engagement remained high and the conversation was on par with previous years!