Middle School Statistics

Tall Tales for Probability Class

Probability is fun.

Nothing drives me nuttier than boring probability units.  Endless worksheets filled with tales of balls in urns and cards being drawn from decks.  Nothing screams for fun and games more than a probability unit.  Here’s some ideas for you to try:


This is a good opener for teaching dependent events.  Take a brown paper lunch bag, and place inside 20 colored cubes.  In the past, I have used 10 white, 6 red and 4 blue.  Do NOT let the students know what chips are in the bag.  Shake the bag, and travel from student to student, having each student draw 3 cubes from the bag (without replacement, or all 3 at once, doesn’t matter).  If they are able to draw exactly 2 whites and a red from the bag, they receive a Jolly Rancher reward.


After every student has had a chance to play, I add an additional challenge to the game.  Letting the class know that there are exactly 20 cubes in the bag, can they predict the distribution of colors in the bag, using the information gathered from the pulls they observed as evidence?  Students who can guess the exact distribution also win the coveted Jolly Ranchers.

Next, dump out the cubes to show the true distribution.  Was the 2 white, 1 red game “winnable”?  Was it possible to win?  Was it plausible that one would win?  I usually try to rig the color distribution so that one can win about 25% of the time.  The 10 white, 6 red, 4 blue configuration will provide a 23.6% probability of winning.


Always remind your students that probability is a long-term ratio.  The 1/6 probability of rolling a five on a die does not mean that every 6th roll will be a five.  In the short term, strange things can happen.

The CBS show “The Amazing Race” provided a great probability teaching moment in its 6th season.  In a “Pit Stop”, teams must complete a challenge in order to earn a clue and move on in the race.  Here’s a summary of the challenge:

In a field are 270 large hay bales.  20 of the bales contain clues.  You must continue to roll out hay bales until you locate one with a clue, at which time you can move on.

Think about these openers:

  • What is the probability you select a bale with a clue on your first attempt?
  • What is the probability you select a bale with a clue on your second attempt?  Third attempt?
  • What is the probability it takes you 5 or more bales to locate a clue?
  • What is the probability you roll out 20 bales, and do not find a clue?

The clip from this show appears below.  You’ll love the reactions of your students!

One last example tale comes from the world of gambling.  While I try to stay away from too much gambling talk in classes, this die-rolling activity leads to a neat backstory.  Have your students roll pairs of dice, and record the number of rolls needed to roll a sum of 7.  After a 7 is rolled, start a new count.  Plotting the results in a class dotplot gives a nice example of a skewed-right distribution, which is often a new shape for our students.

In class, what was the most number of rolls needed?  I have had students get into the 20’s or 30’s.  Is it possible that it could take 50 rolls to get a sum of 7?  Is it plausible?  Note the subtle language lesson happening.

Now, on to craps.  Craps is a fairly complex game, which boils down to this for our example:  once a player rolls, they continue winning until a sum of 7 is rolled, at which point the round ends.

Meet Pat DeMauro:

Craps winner

Pat visited the Borgata casino in Atlantic City in May, 2009 and set the world record for a game of caps, by tossing the dice 154 times without rolling a 7!

Pat’s winning streak was featured in Time Magazine, and features some of the details of the record run.

Probability is exciting!  Make it so!

Algebra Statistics

What’s the Probability That Quadratic Will Factor?

A comment from my post last week about the need for factoring led me to re-visit a question I have posed to classes before, but never allowed to move beyond the “gee, that’s interesting” stage.

Given a polynomial in standard form, with random non-zero* integer parameters a, b and c, what is the probability that the polynomial will factor?

I’ve pursued this question with classes before by writing a polynomial on the board, with blanks or boxes in the a-b-c positions.  Sometimes, I would take “random” shout-outs from the class to fill in the boxes.  With another class, the randint function on a TI calculator was used to generate our abc’s.  The point was to demonstrate that a large majority of quadratics are not factorable, and that despite the nice, rigged, problems we encounter in textbooks, we should spend far more time considering what to do with the messy ones.  But I’d never put pencil to paper and thought about the theoretical probability.

After my post on factoring last week, Jim Doherty mentioned a speaker he had encountered find an experimental probability that a quadratic would factor, and cited 7%.  That number seemed reasonable to me, but perhaps a bit on the high side.  I set up an Excel document to generate three non-zero integers (more on this later), and rigged a system to check for perfect-square discriminants.  I recorded experimental results, in groups of 1000 trials, and kept a running total.

Excel document

Quadratic Graph

After 25,000 trials, I found that 7.26% of the quadratics would factor.

*While this endeavor started off innocently and quickly enough, I had to start over after I realized my Excel document allowed for zeroes.  It took a little logical Excel rigging to exclude them.

So, there must be a theoretical probability out there someplace?  Anyone know how to do it?


Encouraging Persistence Through Contest Problems, Part 2

In my last post, we looked at an AMC-12 problem of moderate difficulty, but with an premise that could be understood by many.  This time, we’ll take a look at a problem which delves into more abstract concepts, and explore how technology can allow students to consider solutions.  The following problem was question #23 from this year’s AMC-12.

Let S be the square one of whose diagonals has endpoints (0.1, 0.7) and (-0.1, -0.7).  A point v = (x,y) is chosen uniformly at random over all real numbers x and y such that x is between 0 and 2012, inclusive, and y is between 0 and 2012, inclusive.  Let T(v) be a translated copy of S centered at v.  What is the probability that the square region determined by T(v) contains exactly two points with integer coordinates in its interior?

In this year’s contest, where over 72,000 students participated, this question was answered correctly by only 4.5% of students, and was left blank by 81.6%.

Tomorrow morning, write this question in its entirety on a side board, and observe student reactions.  How many students begin to sketch the square described in the first sentence?  How many ask questions about some of the sophisticated language?  How many shrug and turn away?  This problem presents a number of chance for students to summarize given information, summarize “scary” language, and consider possibilities.

The first sentence describes a task which all geometry students, regardless of phase or level, can pursue.  Give students the task of telling you everything they can about that square, and share out their ideas.  Let’s look at it piece-by-piece.  First, we have a diagonal with defined endpoints:


Can we find the other diagonal? This is a great opportunity to look at perpendicular bisectors, and consider slope. After the other diagonal is found, we can see the given square:


Is there anything else we might need to know about this square? Can we find its side lengths? How can we find its dimensions? Another nice connection, to our old friend the Pythagorean Theorem, emerges…


How convenient for us! We have a unit square, where all sides have length 1. Even if we don’t consider the rest or the problem, think about how rich of a discussion we have already had!

Now for the scary part….all that spooky language. But is it so bad really? What is the question really asking us to do? Challenge students to re-write the premise of this problem, so that it can be explained easily to a friend or neighbor. Here’s what we are being asked to do:

  • Pick any random point in the first quadrant, but don’t go above 2012 for x or y.  We’ll call it point v.
  • Take the square we just made, and copy it, so that v is the center.
  • How likely is it that the new square contains two points with “integer coordinates”?  This is great time to introduce the term “lattice point”.

Here’s an example of what we are looking at.  The original square remains at the origin, but a new one, with center v1, has also been introduced.  Notice that this new square captures only one lattice point.


So, how are we going to find that wicked probability? Using Geometer’s Sketchpad, I created a model of this problem, which students could then use to explore the premise. Contact me if you would like the Sketchpad file to use.  Enjoy my tinkering in the video below:

Encourage your students to break down problems into smaller, digestible pieces, and not be afraid or scary-looking language. The rich class discussions which come from allowing time for questions to stew and worth it!

So, what’s the answer? Below is a link to a great summary video by Richard Rusczyk from the excellent website It’s perfect for sharing with your classes after you have explored the problem and discussed ideas: