I'm getting ready to plan a problem-based unit on probability and I'm looking for a unit question that can drive our entire investigation. So, this morning I spent about an hour looking online and I was shocked by what I did NOT find. Probability has to be one of the most interesting an relevant topics in high school math and there was not much that caught my attention (and if it doesn't catch my attention, I'm pretty sure it won't catch the students' attention). Here are possible questions that I'm considering right now:

1. The Birthday Problem
What is the probability that, in a room with "n" people, at least two will have the same birthday.

2. Probability Game
I don't know of a great one, but I was thinking about using a "game" to drive our investigation (something like Pig...but preferably more complex)

3. Sports
What is the probability that Team X will make it to the NBA Finals (or something along those lines)

4. Insurance Pricing
Is insurance pricing fair?

The only thing that I really DON'T want to do is Casino/Gambling stuff. Other than that, any suggestion for a unit question is fair game...

p.s. I'm really tempted to create a tab where "we" can start to organize unit questions for reference. Good idea?


03/30/2012 9:08am

Hi Bryan,
I agree that probability is rich with possibilities. Can you share what you want your students to understand about probability by the end of the unit?
I taught middle school, but I always kicked off probability with a 3 question True-False test where I pretended to lose the questions, but asked students to guess "true" or "false" anyway. Everyone played along (they already knew my style). I then gave the answer key and we talked about lots of probability questions. We generated a list of possible responses, then expanded the quiz to more questions, included multiple choice, etc. We developed the Fundamental Counting Principle on our own. Very low tech and low cost, but high return!

03/30/2012 10:15am

I know this probably isn't the response you are looking for, but mostly I just want the students to have a deeper understanding of probability than they do now. In line with what I think you were asking, I hope to talk about calculating probability, possible values and interpretations of probability, calculate probability versus actual results, and (maybe) conditional probability. We have done some combinatorics so they should be comfortable there.

Mostly, I want to use a unit question that will drive all/most of our investigation (so that we are exploring concepts through the context of a central question).

Hope this helps clarify. Thanks for your ideas!

03/31/2012 10:45am

I was curious about a more specific goal for understanding. How would you finish this sentence: "As a result of working on this unit question, students will understand that..."

It might also help to write a sentence about what specific understanding students are bringing. "Students already understand that..."

I've been pushing myself to be very explicit about what I want students to understand, and it has helped me quite a bit in choosing or developing tasks. I wrote about it here: http://www.teachingtweaks.com/wordpress/2012/02/05/make-the-swbat-an-endangered-species/

03/31/2012 1:56pm

@Belinda Your post is great. I was particularly drawn to the subtle, but important, distinction that such a small change can have ("be able to" --> "understand that"). It represents a shift from computation to critical thinking.

In some ways I think you are correct, I need to be more specific about my hopes for the unit. However, I am also hesitant to be too specific. Backwards design gets a lot of hype (and I think there is some validity to the idea) but it also implies that there is a distinct "mathematical trajectory;" that trajectory has been decided upon by the teacher, without regard to the path students take. I hope to plan in a way that is more responsive to the group I am with.

Of course, to a certain extent, our activity each day will still be directed/chosen by me. My hope is that the design of that activity comes in response to the equilibrium that has been reached from the previous lesson, and that the new lesson pushes on that equilibrium for students. This is why, to me, the most important part of the initial planning is choosing a problem/question/task that is rich, perplexing, complex, and interesting to the students. If it meets all of these things, I know we will head in a good direction. I'll wait and see what we "cover" by the end of the unit and assess students on that.

Thanks for your reminder to think things through a bit more...back to the drawing board for me.

03/30/2012 11:47am

Rock, paper, scissors?

Play. The first to win three times? The first to win three in a row?
Is there a strategy that always works? What if you play against a computer? Probability distribution if you play "n" games?

03/31/2012 6:02am

1. Yesterday, a girl in my tiny 9-person class was complaining that she always got put in a group (I usually do 3 groups of 3, and assign them randomly) with either student A or student B. Her group (composed, this time, of her, student A _and_ student B) started calculating the probability of this. As with all probability problems involving combinatorics, this turned out to be complicated!

2. The Monty Hall paradox. This can be fun to simulate.

3. Geometric probability? Maybe not generalizable enough to the other content you'd like them to learn, but see the following famous problems:

a. overlapping schedules: http://www.cut-the-knot.org/Probability/GeometricProbabilities.shtml

b. the stick broken into three pieces, probability it will form a triangle: http://www.cut-the-knot.org/Curriculum/Probability/TriProbability.shtml

c. Buffon's needle: http://www.cut-the-knot.org/Curriculum/Probability/Buffon.shtml
I have never tried to simulate Buffon's needle with a class, but it might be fun, if messy!

4. For conditional probability, there's the famous example of medical testing, where even if you test positive for a disease, there's still a good chance you don't have it. This is because the incidence of the disease in the population is very low, and the test has some nonzero failure rate. (I think there is a discussion of this in "The Drunkard's Walk", by Leonard Mlodinow.)

5. Not the most original suggestion, but Brown's _Advanced Mathematics_, the popular precalc textbook, has some really interesting combinatorics and probability problems. You might be able to take one of those and "glorify" it to the point where it could serve as a launching pad for further investigation.

Have fun!

03/31/2012 10:14am

@Angelo I love the rock, paper, scissors idea...particularly as you begin to ask the probability of winning "x" games in a row

@Mimi Thank you for all of the amazing suggestions! I'm definitely planning on incorporating the Monty Hall problem somewhere. I really really liked the overlapping schedules problem also. You have me thinking...

03/31/2012 10:21pm

I have been thinking about my probability unit coming up (It's a ways down the road, but like you said it is interesting so I can't stop thinking about it). I am basically going to start them off, with a bet. Everyone gets a dei, and we start rolling, we bet which number will win, and combine all our rolls together to see which wins. My idea is to have enough rolls where we have no real winner because they end up nearly equal and hopefully get some ties, so that we have to roll more.

Then I switch it up. We do the same thing but with two dice. Will this change anything?

I would talk about Cardano and how his love of these bets, and gambling is what motivated his development of systematic probability mathematics.

I don't know if this helps, I haven't thought it through fully, but, that's where my thoughts are. Get the students involved in what makes probability so fun. The feeling of luck that pulls us along, but the reality that there is no such thing. Also check out @dandersod's post on the lottery.

04/02/2012 1:08pm

Birthday Problem - One of my favorites. Here's a lesson that I do as part of workshops with middle school teachers. It uses a really cool app.


It would be solid for 11th graders to come up with the formulas that the spreadsheet uses. Why it only takes 23 people in a room to have approx. a 50% chance of a match is usually hard to grasp even after you show them the math especially for many middle school teachers.

Game of Pig - A whole book (http://www.amazon.com/Interactive-Mathematics-Program-Teachers-Guide/dp/1559532521) was written about Pig as part of the IMP project/curriculum (http://www.mathimp.org/general_info/intro.html) Unit 1 for 9th grade. The IMP materials were pathbreaking in project based math.

Another favorite of mine is Buffon's needle experiment - mentioned before. Here's my lesson.


A discussion of why this works would be perfect for 11th grade since it has a nice calculus spin if you want to go there.

04/07/2012 10:08pm

@Ihor: As always, thanks for the thoughtful comments. I looked at the Pig unit in IMP...I like it. Not surprisingly, I really like just about all of the IMP materials. In terms of ready-made curriculum, they come closest to getting students "doing" mathematics of anything I have seen.

04/09/2012 8:26am

I agree. I'm glad you liked it.

04/06/2012 12:38pm

Regarding the Monty Hall Problem, it can be hard to give someone a satisfying explanation to overcome its counter-intuitiveness. A good solution is to change it to a one-hundred-doors problem, where Monty reveals one door at a time, until only yours and one other door remain closed. Now it is obvious why switching your choice improves your odds of winning.
But the MHP should probably come after some more basic probability that does _not_ involve an omniscient, tricky agent (Monty). It's more of a paradox/confuser if the students already understand what a normal one-in-three chance is, and so on.
... Although you don't want to do gambling, perhaps there are other games you can reference. Instead of poker, do they play some kind of collectible/trading card/object game?

04/07/2012 10:06pm

@Matt: Agreed...it can be very tempting with the Monty Hall Problem to just be like, "nope...doesn't matter if you switch." However, I'm less interested with providing a good explanation and more interested in having the students create logical arguments to defend their stance. Maybe this is where the evidence might be uncovered to explain the counter-intuitiveness of the situation?

I loved your trading card idea. I REALLY wanted to do the probability of Pokemon because my students have a serious soft spot for their favorite childhood game. Unfortunately, all I could find was a bit too complex for our tastes and I don't know enough about the game to modify. Maybe next year...

04/09/2012 8:03am

> I REALLY wanted to do the probability of Pokemon...

I've got an idea which might be useful as side comment to them, if not a classroom activity. I think each card has a number on it, such that "127" means that type of Pokemon ("Pincer") is the 127th type published. Suppose someone collected Pokemon regularly over the years, never traded them, and you randomly pulled out 20 cards and checked their numbers. Can you estimate how many different Pokemon types there are, just by the numbers on those cards? According to http://pokemondb.net/pokedex/national , there are 649 types, but you want to know if someone could estimate that from a random sample.

Well, possibly. It is similar to the German Tank Problem (see Wikipedia). In WWII, the Allies wanted to know how many tanks the Germans were producing. The Germans very helpfully painted sequential serial numbers on the side of each tank. By recording a few dozen serial numbers of destroyed/captured tanks, the statisticians were able to use an estimation formula and get (what turned out to be) a far more accurate estimate than the spies, who were using only rumors and informants.

Pokemon is different of course, because there are many copies of each type of card. So you could just suggest that "there's some (Platonic) formula out there that would work." Or, if you're okay with something speculative, you could have them *derive* a formula like so... Start with the German Tank Problem estimator (Wikipedia). Try it on a stack of cards (20-50 cards perhaps). Then look up the total number of Pokemon card types, and see by what factor you are off. This is the "multiple cards per type" factor. Then your new estimator function is the original one, divided by that factor, and you try it on a new stack of Pokemon cards. See how close you get to estimating the true number of card types. Ideally, break the class up into a bunch of groups, and have them each calculate their own factor, and average the factors (tossing out outliers).

(I'm just making this up, and the effect of multiple cards per Pokemon type might not have a linear/multiplier effect. It might be totally bogus exercise, and a misapplication of the estimator. But you can at least tell them about the German Tank Problem, and suggest an estimator for Pokemon cards is out there somewhere, waiting to be discovered. )

04/07/2012 6:07pm

You could play a game that is not fair due to probability, and ask if the game is fair. The students could then create their own games and test the fairness of the game. A good game to start with is rolling two dice with one person getting points for evens and the other points for odd numbers.

04/09/2012 6:28am

A couple of games come to mind.

Have you heard of 1, 2, 3, Shoot!? It's a 3-player variation of RPS where each player holds out 1, 2, or 3 fingers. Player A gets a point if everyone threw the same thing. Player B gets a point if everyone threw something different. And Player C gets a point if there were two same, one different. I think I saw this in Impact Math.

Regarding dice, you could do some interesting math with non-transitive dice. http://en.wikipedia.org/wiki/Nontransitive_dice

I showed students the Monty Hall problem this past fall and after we played it, I had them use spinners to simulate playing the game with a partner 25 times. Each pair would use the strategy of always stay for 25 trials, then they would use the strategy of always switch for 25 trials. Afterward, I collected the class data on the board and we found the overall win percentage for each strategy. With hundreds of trials they themselves conducted, students 1) had to conclude switching was better, and 2) make sense for themselves why it was so. There were a few hold-outs who still thought staying with their first choice was better, but then we had a wonderful discussion where other students changed their minds using mathematical reasoning. I never gave the slightest hint of what the correct resolution was, and students never asked me. THEY had decided for themselves. One of my best lessons ever.

05/04/2012 8:14am

I posted a probability question online for high school students that incorporates many elements. It involves removing colored balls from a bag. I think the link works if you click on my name above. Feel free to use any or all of it.

05/04/2012 8:17am

The link is there, but it got a bit mangled.


05/22/2012 3:35pm

The website I provided here is a link to my blog, and a problem which appeared on this year's American Math Competition. I like the problem because it has a fairly simple premise to explore. But, it does not have a real-life context, which should really be the theme of any probability unit.

Another problem I like is the cereal box problem. Here's a nice site for it: http://mste.illinois.edu/reese/cereal/


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