**Previously**

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.

**Act One**

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. Itâ€™s more productive and more interesting even though its easier to empathize with the buyer initially.

**Act Two**

Ask: “What information would you need to decide on a fair price?”

Now weâ€™re going to introduce the probability distribution model.

Itâ€™s unusual so weâ€™re going to do several things in order:

- Weâ€™re going to ask for speculation about what it means. Then weâ€™re going to tell them what it means. (A: It shows every possible event in the space of events along with their likelihoods.)
- Weâ€™re going to show more contrasting cases. (See: Schwartz, 2011.) Impossible cases and possible cases. Weâ€™ll ask them which are impossible and why. Then weâ€™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 weâ€™ll ask the students, “If youâ€™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 youâ€™re cheaping your customers and eventually no one will buy your ducks. The second distribution means youâ€™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 wouldnâ€™t result in money lost or gained.

After laying all this informal groundwork, weâ€™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.

**Act Three**

Thereâ€™s no act three here. We donâ€™t know the probability distribution of the Money Duck (I asked) so we canâ€™t validate. Thatâ€™s okay.

**Sequel**

Letâ€™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?”

## 4 Comments

## Robert Kaplinsky

May 14, 2014 - 9:21 am -This is a great result of the collective effort you led. You should send a link to this page to the company you asked for the information and see if they’ll give you a response or comment about how this process compared to their own.

## Bowen Kerins

May 14, 2014 - 7:13 pm -One other bummer-world version is the Wheel of Fortune slot machine found in Vegas.

On inspection, the giant wheel at the top of the game has some great value: there’s a $1000 space, a $500 space, some others.

And you can do some calculations, figure out how often the bonus is activated, and learn that based on what you see, you should come out ahead if you keep playing the game.

But of course you don’t, because it’s a slot machine, and the wheel is rigged. Vegas actually has laws that limit precisely how rigged the wheel is allowed to be. The nature of the rigging has led to this lawsuit in Michigan: http://www.rgtonline.com/article/lawsuit-alleges-wheel-of-fortune-slot-machines-are-rigged-24448

If by “bummer-world” you meant “math textbooks”, I was able to work some of these kinds of problems into the CME Project books, but they had to be carnival games and not gambling games. One example here: http://imgur.com/gw1k43i

## blaw0013

May 31, 2014 - 3:41 pm -While reading, I thought about how I might go about trying to resolve some of the questions posed in relation to this problem. I was reminded of a surprising technique to count/estimate unknown amounts — in the case of the youtube link the context is estimating the number of taxi cabs in London. Enjoy.

http://youtu.be/tyX79mPm2xY