# Dan Meyer’s Money Duck problem part 2

Last week I wrote about an interesting expected value / probability question that Dan Meyer posed on his blog.  Dan’s blog post is here:

[Confab] Money Duck

My first post about the problem is here:

https://mikesmathpage.wordpress.com/2014/05/05/dan-meyers-money-duck-vs-pepsis-play-for-a-billion/

In the comment section on Dan’s blog I described an exercise that I thought would be fun to try out with kids:

Your friend Dan walks in with 5 money ducks. Each of the ducks has some money hidden inside – one has \$1, one has \$5, one has \$10, one has \$20, and one has \$50. You do not know which duck has what amount of money, but Dan does.

He suggests a couple of games:

(1) You pick any of the ducks you want and get to keep the money inside of it. How much money do you expect to win playing this game one time?

(2) You pick any of the ducks you want and then Dan tells you the lowest amount of money remaining in one of the 4 ducks that you didn’t pick (without telling you which one of the ducks contains that lowest amount of money). He then lets you pick a new duck if you want. Would you rather play this game or game (1)? Why?

(3) Same as (2), but after your first selection Dan tells you the highest amount of money that remains in one of the four ducks that you didn’t select instead of the lowest. He then lets you pick a new duck if you want. Would you rather play this game or game (2)? Why?

Depending on how advanced the students are, you could ask for expected values in games (2) and (3), too, but even without calculating an exact number, the discussion about which games people would rather play would be interesting.

Today I finally got around to playing these three games with my kids.  Here’s how it went.

First, since the my original blog post about this question mentioned the similarity between Pepsi’s “Play for a Billion” game and the activity that many people commenting on Dan’s blog wanted to try out with the Money Ducks, I brought out a couple of the old props from the show to help introduce the problem:

Now, on to the game.  The first game will be a one-shot game where you roll a 10-sided die and you have an equal chance of winning \$1, \$5, \$10, \$20, or \$50.  The question is – how much money do you think you’ll win when you play this game?  Interesting to hear the boys trying to reason out the expected value:

At the end of the last video my older son thought we could figure out the expected value by playing the game 100 times.  We didn’t want to actually do that, but we did go through that math and concluded the expected value for playing the game once is \$17.20.   At the end of this video I tried to explain that you won less than \$17.20 60% of the time and more than \$17.20 40% of the time.  I was trying to make the point that just because \$17.20 was the average amount of money that you won, that doesn’t necessarily mean that you win more than that half the time and less than that half the time.  I did not explain this point very well at all 😦

Next we moved on to the two more complicated games that I thought would be fun to play.  In this video we play the game where the prizes are hidden, but I give you a little extra information after you select one of the prizes.    After you pick, and before you learn what your prize is, I tell you the lowest amount of money that you didn’t pick and then see if you want to pick a new prize.  Although analyzing this game isn’t super complicated, I didn’t want to go into the details.  Instead we just tried to talk through whether or not they thought you’d win more money playing this game or the first game:

The final game we played is similar to the one in the previous video.  The only difference is that after you pick your prize, I reveal the highest amount of money remaining from the ducks you didn’t pick.  It was fun to hear how the boys talked through whether or not they would want to play this game or the prior one.   Again, a complete analysis of this game isn’t super complicated, but the goal here wasn’t to do a detailed analysis, but rather to simply talking through the ideas.

All in all, I think there are lots of fun and mathy conversations that you could have with kids about the Money Duck.  As I wrote in the prior blog post, I think the “how much would you pay for the Money Duck” question is a little more complicated than just expected value, but it is still a fun question.   Thanks to Dan for asking for thoughts about fun ways to play with the Money Duck – it was fun to think through.