Many, many thanks to everyone who helped me sort out some thoughts on this lesson in our previous confab post, including but not limited to Fawn Nguyen, Robert Kaplinsky, Bowen Kerins, Dan Anderson, and Max Ray, and the many participants I pestered at OAME2014 this last week.
Here’s the download link at 101questions.
Show this video.
Ask: “What would be a fair price for the Money Duck?”
You guys were right. In the end it makes more sense to pose the student as the seller. It’s more productive and more interesting even though its easier to empathize with the buyer initially.
Ask: “What information would you need to decide on a fair price?”
Now we’re going to introduce the probability distribution model.
It’s unusual so we’re going to do several things in order:
- We’re going to ask for speculation about what it means. Then we’re going to tell them what it means. (A: It shows every possible event in the space of events along with their likelihoods.)
- We’re going to show more contrasting cases. (See: Schwartz, 2011.) Impossible cases and possible cases. We’ll ask them which are impossible and why. Then we’ll tell them which are impossible and why. (A: The probabilities have to add up to 1. Each rectangle here has to stack up and reach exactly the 1 line.)
- Now we’ll ask the students, “If you’re selling the Money Ducks for $5, why is each of these distributions bad for business.” (A: The first distribution means word gets out that you’re cheaping your customers and eventually no one will buy your ducks. The second distribution means you’re losing loads of money.)
- Show the students four distributions and ask them to make up a price for the distributions that would be fair to both buyer and seller, that wouldn’t result in money lost or gained.
After laying all this informal groundwork, we’re ready to transition from qualitative descriptions to numerical and define expected value.
- We ask them to calculate the expected value of the distributions from #4 and compare those values to their prices. If their intuitions were sound and their calculations correct, their intuition will support the validity of the expected value model.
There’s no act three here. We don’t know the probability distribution of the Money Duck (I asked) so we can’t validate. That’s okay.
Let’s show the students the actual price of the Money Duck and ask them to determine a probability distribution that would give them a $3 profit per Money Duck as a seller. Answers, happily, can vary.
BTW. The bummer-world version of this problem reads like this:
A carnival game is played as follows: You pay $2 to draw a card from an ordinary deck of 52 playing cards. If you draw an ace, you win $5. You win $3 if you draw a face card (Jack, Queen, King) and $10 if you draw the seven of spades. If you pick anything else, you lose your $2. On average, how much money can the operater expect to make per customer?
2014 May 12. You should definitely read Dan Anderson’s experience running this lesson with students.
2014 May 19. Also, Megan Schmidt had some interesting results with students particularly w/r/t the question, “Which distributions are impossible?”