In my modeling workshops this summer, we first modeled the money duck, asking ourselves, what would be a fair price for some money buried inside a soap shaped like a duck? We learned how to use the probability distribution model and define its expected value. We developed the question of expected value before answering it.

Then the blogosphere’s intrepid Clayton Edwards extracted an answer from the manufacturers of the duck, which gave us all some resolution. For every lot of 300 ducks, the Virginia Candle Company includes one $50, one $20, one $10, one $5, and the rest are all $1. That’s an expected value of $1.27, netting them a neat $9.72 profit per duck on average.

That’s a pretty favorable distribution:

They’re only able to get away with that distribution because competition in the animal-shaped cash-containing soap marketplace is pretty thin.

So after developing the question and answering the question, we then *extended* the question. I had every group decide on a) an animal, b) a distribution of cash, c) a price, and put all that on the front wall of the classroom â€“ our marketplace. They submitted all of that information into a Google form also, along with their rationale for their distribution.

Then I told everybody they could buy any three animals they wanted. Or they could buy the same animal three times. (They couldn’t buy their own animals, though.) They wrote their names on each sheet to signal their purchase. Then they added that information to another Google form.

Given enough time, customers could presumably calculate the expected values of every product in the marketplace and make really informed decisions. But I only allowed a few minutes for the purchasing phase. This forced everyone to judge the distribution against price on the level of intuition only.

During the production and marketing phase, people were practicing with a purpose. Groups tweaked their probability distributions and recalculated expected value over and over again. The creativity of some groups blew my hair back. This one sticks out:

Look at the price! Look at the distribution! You’ll walk away a winner over half the time, a fact that their marketing department makes sure you don’t miss. And yet their expected profit is *positive*. Over time, they’ll bleed you dry. Sneaky Panda!

I took both spreadsheets and carved them up. Here is a graph of the number of customers a store had against how much they marked up their animal.

Look at that downward trend! Even though customers didn’t have enough time to calculate markup exactly, their intuition guided them fairly well. Question here: which point would you most like to be? (Realization here: a store’s profit is the area of the rectangle formed around the diagonal that runs from the origin to the store’s point. Sick.)

So in the *mathematical* world, because all the businesses had given themselves positive expected profit, the *customers* could all expect negative profit. The best purchase was no purchase. Javier won by losing the least. He was down only $1.17 all told.

But in the real world, chance plays its hand also. I asked Twitter to help me rig up a simulator (thanks, Ben Hicks) and we found the *actual* profit. Deborah walked away with $8.52 because she hit an outside chance just right.

Profit Penguin was the winning store for both expected and actual profit.

Keep the concept simple and make winning $10s and $20s fairly regular to entice buyers. All bills â€“ coins are for babies!

So there.

We’ve talked already about *developing the question* and *answering the question*. Daniel Willingham writes that we spend too little time on the former and too much time rushing to the latter. I illustrated those two phases previously. We could reasonably call this post: extending the question.

To extend a question, I find it generally helpful to a) flip a question around, swapping the knowns and unknowns, and b) ask students to *create* a question. I just hadn’t expected the *combination* of the two approaches to bear so much fruit.

I’ve probably left a lot of territory unexplored here. If you teach stats, you should double-team this one with the economics teacher and let me know how it goes.

*This is a series about “developing the question” in math class.*

## 7 Comments

## Kyle Pearce

September 11, 2014 - 7:22 am -Love this task!

I used to teach a grade 12 stats course and this would have really livened up our unit on probability distributions.

You challenged us at OAME 2014 in Toronto to come up with some way to use that money duck; although most said “expected value,” I don’t think anyone there could see it developing into such a rich task.

Thanks for the share. Just sent on to some stats teachers in my board.

## Robert Kaplinsky

September 12, 2014 - 4:37 am -Wow! I had no idea someone actually was able to get the information on the money in the money duck. Well done Clayton! That is one critical piece of information that takes this task to the next level. Very cool.

## Ian D

September 12, 2014 - 2:42 pm -Nice post!

For the actual profit how many customers were you using in the simulator?

It’s also good to consider what happens if the customers get hooked on buying ducks and then we could consider the time to ruin for an initial stake of say 2 ducks, where the customer reinvests the profits into the duck gambling enterprise. Some questions could be:

1. What is the expected time for a customer to go bust?

2. What is the median time to go bust?

3. For 10,000 customers what is the maximum time that it takes a customer to go bust?

For the duck company to be like a casino it would like a high expected time to ruin (to entice the customers), but a low median (to milk most of them dry quickly). Some quick simulations from 10000 from a quickly written bit of code:

Let’s assume that ruin is where the amount of money owned is negative, given a customer starts with the money for two ducks.

Real duck company:

expected time to ruin 3.093

median 3

maximum time playing 14

Panda

expected time to ruin 68.03

median 7

maximum playing time 13185

Penguin

expected time to ruin 6.513

median 3

maximum playing time 101

So, Panda would probably do quite well providing it can make enough ducks to sell to the very few individuals who get lucky. There is perhaps some danger that the Panda company may get taken themselves to the cleaners. I think I would prefer choose to set up like the Penguin company.

## Dan Meyer

September 13, 2014 - 6:49 am -Love the introduction of time.

There were about 40 customers who made three purchases each. So we aren’t yet close to the “expected” part of “expected value.” Lots could go wrong.

## Julie

September 14, 2014 - 8:42 pm -What a fun way to engage your students and have them learn by doing! Reading this blog post made me want to sit down and write up a business model of my own. We all know that getting students involved is the best way to help them learn. Kudos! You’re doing a great job.

## Gilbert

October 1, 2014 - 4:46 pm -You could go even deeper on this question with behavioral economics. Why is everyone behaving irrationally in the way they are?

Your setup isn’t all that different from Kahneman and Tversky’s classic prospect theory studies.