Urban Expectation Values

Goal

Students calculate expectation values in the dice game Machi Koro.

Note

This is a longer variant of the lesson using the Armadillo game.

Time

90’

Core Concepts

  • discrete event

  • probability mass functions

  • uniform distribution

  • triangular distribution

  • expectation value

  • histogram

The Game: Machi Koro

  • the game works in 4 teams of 2, so 8 people can play at one table. So 2-3 boxes are enough to keep most classes busy.

Hint

For a shorter game and explanation time in Machi Koro, remove the fourth player building. Also remove the purple buildings.

How to use this Lesson

This lesson introduces discrete events and probability mass functions. Rolling dice is a proper way to sample enough data in a short time. Instead of enumerating a lot of different distributions, you analyze the uniform and triangular distribution more deeply. On the way, cover all the concepts, including plots of the two distributions. The Machi Koro game gives your students a clear incentive to calculate expectation values properly.

Hint

If you’d like to introduce expectation values with a simpler game, take a look at penguin_poker.

Lesson Plan

  • play a round of Machi Koro or a similar game

  • stop the game when all tables have finished or after a 20’-30’ time box.

  • now you want to analyze the total worth of one players’ city.

  • introduce the concept and equation for the expectation value.

  • calculate one city for a roll of one die together on the board

  • do the same for a roll of two dice.

  • you can introduce assumptions for handling the cafes: in the early phase of the game it is easy to spend most of your money. So you might assume that you do not pay anything.

  • in the phase where you are saving up for a big building the cafes become quite important. You might assume that you always pay the full amount to the other cafe owners.

  • with cafes you may find a negative expectation values

Extra tasks:

The alternative rolls with one or two dice are a great opportunity to introduce histograms and distributions of discrete variables.

Things you can do:

  • ask students to draw a histogram of one dice roll (the probability mass function is a discrete uniform distribution)

  • ask for the probabilities of two dice rolls. Write them on the board as well.

  • ask studens to draw a histogram of two dice rolls (a triangular distribution)

  • students might also figure out the number of combinations for three dice as well. It is a bit tedious to go through all the permutations, but it is totally worth it. When they plot the histogram they might start regonizing something similar to the bell curve. In the end, you get an opportunity to introduce the central limit theorem, which can be explored further in an extra session on distributions.

Reflection Questions

  • how is the expectation value defined?

  • is it better to get 10 coins on a roll of 7 or 12 coins on a roll of 6 with two dice?

  • in what situation could you have a negative expectation value?

  • how does a histogram differ from a bar plot?

  • what is a good winning strategy in Machi Koro?