MathNotations

While you're celebrating New Year's Eve (meaning you're probably not reading this blog!), or thinking about your favorite math teacher (and probably keeping it to yourself), or considering clicking on the new subscriber chiclets in the sidebar, I thought I would kick off 2008 with something different.

As we were playing family Bingo a few days ago with about three dozen families (my wife was reluctant to go and, of course, she won the first game!), I began thinking about the underlying mathematics of the game and its many variations. I know what my wife would be thinking: "Dave, Why can't you just enjoy the game without analyzing it!"

Here were some thoughts running around in my head, when I should have been concentrating on the two boards I was playing (I won nothing BTW):
(1) Historically: What are the origins of this game? Designed by some brilliant mathematician or was it just some game of chance that evolved?
(2) The number of possible boards must be astronomical. How many different boards are supplied by the companies that manufacture this product?
(3) If I buy a bingo game that comes with, say, 36 boards, are these same 36 boards in every box? If I buy a set from a different manufacturer will the boards overlap or be entirely different?
(4) Are all boards randomly generated by software these days? If a bingo game is to be played in a large hall, with hundreds or even thousands of players, how likely is it that there will be multiple winners in a single game? Do the boards all have different winning lines or are there lots of overlap among boards?

Some probability thoughts:
(5) What is the probability of a winner after the minimum number of balls drawn from the bingo cage, namely four numbers (don't forget the free space!). Since I've never seen this happen, I'm assuming the chances are virtually zero!
(6) More realistically, for about two dozen players, each playing a single board (or one player using 24 boards!), what the expected number of balls drawn before a winner occurs? I was conjecturing less than half of the seventy-five numbers, maybe low thirties.
(7) How did the probabilities change as one increases players and boards? I assumed many mathematicians had already solved all of the intricacies regarding the probability of a winner after 10 numbers, 20 numbers, 30 numbers, etc. Instinctively, I felt that this was a very sophisticated problem, probably beyond my comprehension, but I wanted to know more.

Of course, when we returned home, I did some online research of the game -- fascinating stuff: Origins in Italy, Lotto, Beano, Bingo, Mr. Lowe (the toy manufacturer), Professor Leffler from Columbia University, the fund-raising aspect that started in a church in Wilkes-Barre, PA, and so on...
You can easily find these same sources so I'll leave that for our readers. However, there was a dearth of serious mathematical analysis of the probabilities and the combinatorial aspects. I only found a couple of these and neither went into much explanation of the underlying theory, other than to suggest it is complicated, oh, and data tables generated by some software. Of course, I'm sure I missed some wonderful references that my readers will find.

So I decided to do what I usually do when facing a complicated task (a la Polya): Reduce it to a much simpler problem! Not only to understand it better for myself, but, in the back of my mind, I was thinking of how a middle schooler could begin to understand the complexities of all this.

What could be easier than a 2x2 board - just 4 little numbers on a card and to really oversimplify it, only four numbers will be available: 1,2; 3,4. The semicolon separates the possible values for the first column on the card from the 2nd column. I will use this notation from now on. So here's the first elementary question for the reader and for the student:

STUDENT/READER QUESTION #1:
Assume there is one player with one card using the numbers above. Explain why the probability of winning this simple 2x2 version after two numbers are called is 1, that is, 100%.
You're thinking: Way too obvious a place to start, right! Too boring for the student...

STUDENT/READER QUESTION #2:
Ok, let's dial it up a tad. We'll still keep it a 2x2 card, but, this time, there are three numbers available in each column: 1,2,3; 4,5,6. Remember this notation means that 1,2,3 are the possibilities for the first column and so on.
Again, one player with one card: What is the probability of a win after two numbers are called?
Comment: There are many methods here from listing all of the possibilities to permutations and combinations to multiplication of probabilities (one number at a time without replacement), etc. I believe, pedagogically, it is important for the student to see more than one way!

I could stay with the 2x2 game and add more numbers but the student and our readers are an impatient lot and want to move on to something more interesting, right? So let's move on to a 3x3 board which is much more like the 5x5 board in that it has a free square in the middle. But we have to start slowly here - trust me!

STUDENT/READER QUESTION #3:
Now we have a 3x3 board. The available numbers will be simply 1,2,3; 4,5,6; 7,8,9. Again, one player, one card. Couldn't be easier, right?
(a) What is the probability of a win after TWO numbers are called? That's the minimum number with the free space covered.
(b) A little harder now: What is the probability of a win after THREE numbers are called?

Ok, we'll ask the same two questions with more numbers available:
Suppose the possible numbers are: 1-6; 7-12; 13-18


(c) Now, what is the probability of a win after TWO numbers are called?
(d) What is the probability of a win after THREE numbers are called?

I better stop here! This is enough for Part I. As usual, any results I've stated need to be verified by my readers and don't forget to give proper attribution if using any of this in a classroom setting.

Posted by Dave Marain at 6:29 AM 10 comments

Labels: Bingo, combinatorial math, investigations, middle school math