How I teach random variables and expected value

Games of chance are a great way to engage students in probability and statistics. The topic of expected value is a great way to teach a life lesson: don’t gamble. Here’s the low-down on my unit on probability distributions and expected value.

I start with an introductory activity where the students learn how to do a weighted average (which is essentially what the calculation of expected value is). During the pandemic, I turned this into a pretty sweet Desmos activity (DESMOS: Weighted Average) where students estimate the final grades of 3 students in a math class before calculating the actual averages. I took a play out of Steve Phelps’ playbook to aggregate the students guesses on a dot plot. Here’s the google sheet for the students to calculate the weighted average.

Next, I teach the students a little vocabulary and the key concepts through a short series of worked out examples (about 15 minutes). Here’s the note guide I use. (Download below)

Here’s another Desmos activity (DESMOS: Expected Value Practice) that is self-checked which gives the students a little bit of practice on the concepts that we just learned. I love the feedback feature so that I can give the students feedback real-time.

Next, the students do a little probability experiment with three dice and do some more work calculating and interpreting expected value. Since we were remote for the last week, I had them use random.org to roll 3 dice to take their data. (DESMOS: Additional Expected Value Practice)

We do several practice problems involving lotteries and casino games to show them that ALL lotteries and casino games have negative expected values.

GREED – The Expected Value Game

Finally, we play a game called GREED.

This game is one of most memorable activities of the year for some of the students. Each student stands next to a chair. I roll a six-sided die. If I roll a 5, students get five points. If I roll a 3, students get three points…or whatever number I roll as long as they are still standing. Since I have whiteboards around my room, the students tally their points at the board. They continue tallying points EXCEPT if I roll a one; then, any student who is still standing will lose all the points that they’ve tallied for that round. A student can save their points by taking a seat between rolls, but once they’ve sat down, they can’t stand back up that round. A round ends when a one is rolled or all students are seated. At the end of the round, all students count their tallies and save their points by adding it to the previously accumulated points. Don’t announce that it will be “last round” because desperate students will just stay standing until they win or they bust.

The trick of the game is for students to decide whether the cost of losing all the points they’ve accumulated during a round outweighs the benefit of gaining more points. This is a perfect way for students to get a sense for expected value. Obviously in this game, the expected value of your gain changes as you accumulate more points. So what is the “sweet spot”? When should you take a seat? It turns out that when you get to 20 points, the expected value of your gain is zero, so generally, you should take a seat once you’ve gotten to 20.

Expected value is a fun topic to teach. I hope some of the ideas and activities are helpful.