# Card Auction (Introducing dependence)

Last month, Nat Banting described a fantastic task on his website called the Dice Auction. You should really just read the original post, but I will summarize it as best as I can here.

The premise is that you are invited to an auction, and given a budget of \$10 [I changed the budget to \$15 for my students to encourage a bit more risk taking]. Everyone at the auction has the same budget. The participants are all bidding on certain events that may occur when two 6-sided dice are rolled (e.g., both numbers are greater or equal to 5; a single 2 is rolled; both numbers are odd; etc.). After all the events have been auctioned off to the highest bidders, the two dice are rolled 20 times. Each time the event that you purchased occurs, you collect a prize. Bidding always begins at \$1 and goes up in increments of \$1. You cannot bid against yourself. The order of the events up for auction is known beforehand. If you choose not to spend (some, or all of) your money, the auctioneer will sell you prizes at a cost of \$2 per prize after the bidding has ended. Your task is to get as many prizes as possible.

A couple of weeks ago, I tried this activity in my math club. Like Nat, I used pattern blocks as currency and brought candy for prizes. Each student received a handout with the description of the events up for auction (you can see Nat’s original handout on his website; I deleted 5 squares because of limited time and because my group is much smaller). I gave the students a couple of minutes to think about which events they wanted to bid on, suggesting that they didn’t share their reasoning at that stage. I didn’t suggest that the students calculate anything, or even mention the word probability; like Nat, I was interested in tapping into their intuition.

Once the auction began, the students bid eagerly, but clearly strategically. (One student did spend a lot of his money on the first square, “both numbers  5″, which he regretted very shortly after, and there were a few squares that sold for much less than I expected.) Each item opened at \$1, and I collected money after each event closed. Bidding went pretty quickly, but still a bit longer than I had hoped, even with a small number of students. (The problem for me is that I only have one lunch hour with these students per week, so time is a precious resource.) After all of the events were sold, the dice were rolled 20 times, and prizes were distributed after each roll. The students kept track of how many times their purchased events were rolled (in the tally section of the handout) so that we could combine the results at the end and discuss the frequencies with which they occurred.

The great thing about this activity is the conversations that happen afterwards. Who “won”? Was there an event that occurred more often than expected? Less often? Did someone pay too much? Too little? Is there a square you regret paying for? (The student who spent much of his money on “both numbers  5″ certainly did.) Which events would you bid on next time?

These students were more comfortable with the basic concepts of probability than I expected, so most of their arguments were mathematical (“Well, there are x different ways of getting…”). We discussed the difference between theoretical probability and experimental probability – or rather, why there was a difference between the two. (I was surprised when one of my students brought up the Gambler’s Fallacy in her explanation – she’s clearly been doing some extra-curricular reading!) Unfortunately, the discussion didn’t last as long as I had hoped, and I didn’t have the luxury of carrying it over to the next day. Because my students loved the activity, and because I wanted to squeeze more mathematics out of it, I decided I would do it again – but with a twist.

It was clear to me that the students were very comfortable with the notion of independence and with calculating the probability of independent events. So, to make the task more challenging, I decided to switch from dice to cards. The premise of the auction is exactly the same (see above) – however, instead of two dice being rolled on each turn, two cards are drawn from two separate decks. The decks are not shuffled together (although this is certainly a valid variation), and the cards are not replaced. (Of course, you can also introduce dependence with dice, which would necessitate that you keep track of previous rolls. Creating suitable, not-too-unlikely events is a bit more challenging of a task in this case.)

I introduced the idea to my students last week. After we had a discussion about what was different about the two auctions, they suggested a few possible events that could be auctioned off (which had probabilities strictly between 0 and 1), and we had some great conversations about the probabilities of these events on the first draw, on the second draw, and so on. Since some of these events involved sums and products, we decided on a few rules for the Ace, the face cards and the Joker: in particular, the Ace has a value of 1, the face cards (Jack, Queen, and King) all have a value of 10 (to simplify the calculations of sums and products), and the Joker has a value of 0. You and your students may come up with different rules for these cards, which would change the probabilities for certain events – another interesting topic of conversation!

Again, time was not on our side, so I took the students’ suggestions but needed to add a few events myself. I wanted to make sure that the board was “balanced” in that there weren’t too many very likely and very unlikely events, so a friend and I played around (at a pub, over drinks – is there a better way to do math?) with a lot of different possibilities until we got a well-rounded board. To simplify our calculations, we only considered probabilities for the first draw, but of course, you would want to talk with your students (as I will with mine) about what happens in subsequent draws. The calculations do get unwieldy, though, so if you are interested in finding exact probabilities (e.g., for the first 1, 2, or 5 draws), I would choose only a few events from the board to focus on. There are several on here that could generate some great discussions and interesting solution methods (e.g., sum is odd; product less than 10, sum less than or equal to 9). For students new to probability, however, I imagine this being used as an introduction activity to the notion of dependence, leading to interesting discussions and giving just a taste of the calculations involved.

Below is the board that I will be using with my students (link to handout here). I have narrowed it down to 16 events and will be making only 15 draws because of limited time and a small number of students; for a larger group, you will want to add events to give everyone the opportunity to participate. As I suggested above, the activity is infinitely adaptable – not only in the events that you (and your students) choose, but also the rules that you assign to the various cards, how/whether you shuffle them, how many decks you use… If you have any suggestions for adaptations, comment below or on Nat’s original post (and do check out his other tasks, if you haven’t already – his blog is a treasure trove of interesting activities and projects).