For the last three weeks I’ve been studying probability with my Algebra 2 students, thinking: “too bad there’s no good way to teach this.” Probability only works on large scales, and then it really only *approaches* working. The kids all get the basic stuff, like coin flips having 50/50 odds, they all don’t get the more complicated stuff like standard deviation. They lack the tools for problems with continuous distribution, and as far as I can tell that leaves us with carnival games and card tricks. Which only work on average. By being my interested, lively self during class I managed to interest half of them for most of the time (what are the odds that you’re interested in this question, Johnny?). But it was a hard unit for me.

Today, last day of class, the final is over, and I think to myself, “ok, let’s talk about the Monty Hall problem.” (Knowing what the problem is is necessary to understanding this post.) I bring the requisite bowls and candy prizes.

“Who wants to play?” I ask, and J raises her hand. They don’t know the rules yet. I just say, “One of these three doors (bowls) has a tootsie-roll pop in it. If you guess which one, you get to keep the candy!”

So she guesses, and gets it right, and I give her the lolly pop, arcing it to her desk.

“Who else wants to play?” More hands go up, and we play several more times with this simple rule. ”What’s the probability of winning?” I ask, and they easily respond with some form of one third.

On the fifth round, L is playing, and she makes her choice. I say, “L, your guess was as good as any other. But I want to give you another chance. I will show you that this bowl over here <lifting bowl> is empty!” She, and the rest of the class, stare at me blankly. ”Do you want to keep your original guess, or switch!?” I say this channeling Regis Philbin on the $500,000 question. She stays. ”Is that your final answer?” She says that it is, and with a flourish I show her that she is right; she’s won the candy! Congratulations! Congratulations all around!

We play again. I stop after showing an empty bowl to ask about the odds of the last two unknown bowls, and the entire class is quite confident that there is a 50% chance that the candy is under bowl 1, and a 50% chance that the candy is under bowl 2. This is intuitively obvious and not true.

Excellent.

They still don’t see anything fishy when we play another time, and again I ask about the chances of the bowls. I say, “So, when you chose at first, you had a 1/3 chance of winning. Now you say that the same bowl you picked has a 1/2 chance of winning. How could flipping over an empty bowl over *here *improve the chances that *this* bowl is a winner?” The students sort of stammer – they aren’t fluent in this language and don’t have the vocabulary to convince me that it’s true. A student in the back starts to give an example.

He says, “Imagine if you had fifty bowls,” and I immediately whirl around and draw 50 bowls on the board. It takes a long time, and I make a spectacle out of it.

“Ok,” he continues, a little incredulous that I actually just drew fifty marks on the board. ”And the player chooses one, so -”

“Which one?” I interrupt.

“Uhh…”

“Please come up and circle one!”

I say, “What are the odds that you just chose the right bowl?”

He says, “1/50,” and the rest of the class agrees.

I say, “Ok. So now, as the host, I’m going to open all the doors but the one you chose and one other. If you chose right at the beginning, the extra door will be empty. If you chose wrong, it will have the candy in it. Here we go!” And I make a big, slow deal of erasing all but one other mark.

“Which bowl do you think has the candy?” And they all think that the lower left bowl probably has the candy. Several students are laughing at this point.

“What are the odds?” I ask, and they falter!

“Aren’t they 50/50?” I ask. ”There are two things, and one of them has the candy, and we don’t know which one, right? So what’s the big deal?”

H protests, “but in this example you *chose* the right one to leave! Obviously it has the candy!”

“Isn’t that what I did before?”

The rest of the lesson goes on like you’d expect. We make a tree diagram. I extend the example: imagine you had to pick the correct blade of grass out on the lawn, and then I went and mowed all the other blades of grass except one. Do you think you’d pick correctly first, or second? Etc. We work out numerical probabilities.

But this lesson is different from the others in probability in that the kids are engaged. This problem should have been first. I wish I based the whole subject of probability around it:

- The students can describe the problem well. They think they are fluent at first but later find they are not, and are intrigued to find better ways to describe what is going on. No other probability scenario I found had this quality.
- We don’t have to try a million times before the probability gives actual results. Because of the role of the host, knowing probability actually helps kids win the game in an interesting way (much better than “you should bet on 7 because it comes up 1/6 of the time which is more than other numbers”).
- This game is actually fun to play. Who knows why. The switching thing is great. All of my quarter and die games were flops. For the last five years.

I thought I would like teaching probability, but I don’t, because it claims to be so practical and is actually so impractical. Do you have any ways of teaching it that you like?

This example is so practical, it’s borderline too scary to use in class.

From my notes this term:

This is straight from “Calculated Risks” by Gerd Gigerenzer. He polled a bunch of doctors (most or all of whom were heads of their departments), and they had NO CLUE how to answer this. They guessed high, like 90%. The actual math works out to about a 9% chance of

actually having cancer.In other words, this stuff is worth knowing well because ‘experts’ may not know how to think it through on your behalf.

Found out about the book from here:

http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/

Oh, that said, it was still far more fun to use Dan Meyer’s poker tournament clip and analyze that.

http://blog.mrmeyer.com/?p=3243

I just finished teaching probability, and I’m sad to say that I don’t have any additional ideas.

I will share an activity that drew connections for some of my students. We flipped cups to see if they landed on their side or on end (either end). Since our cups were tall and skinny, they landed on their side significantly more often (86% if memory serves me correctly), which was roughly how much larger the surface area of the side was compared to the surface area of the ends (about 86% of the cup’s surface area was located on the side of the cup). The kids thoughts that was interesting, and they enjoyed conducting the experiments with cup tosses.

There was a big hoopla about this type of problem when Marilyn vos Savant answered the question about a game show in Parade magaine. The story can be found here:

http://www.marilynvossavant.com/articles/gameshow_print.html?t=64

With some good story telling, this would be an equally good lesson.

I never understood before why the situation with 50 doors helped people understand the problem. I think I get that better now. Thanks.

Teaching probability seems so impractical because it is wrapped in hypotheticals.

Instead of telling them about the game of roulette, spend some time playing it and get the students to deconstruct the game: what are each of the probabilities and how do they compare to the payouts? Move to a harder game like craps. Finally, ask the students to create their own games and have your own Casino Night. At the end of the night, ask the students to analyze their winnings/losses. What does it mean about the design of the game?

Our fifth annual Casino Night is coming up tomorrow night. This may deserve a blog post of its own…

Teaching probability seems so impractical because it is wrapped in hypotheticals.I think you’ve hit it here exactly – the whole introduction to this post in a single sentence!I tried a casino-style thing last year, but was disappointed with the games the students created. It was worse this year without the games, so I probably need to think about a better structure.

@Sue: It’s much better on the actual board, where you can slowly wipe away all of the other ones. It becomes VERY clear that the prize will be in the other bowl.

@josh G – I like that example because the results are so unexpected. I used a similar one this year but the class wasn’t very engaged. What I’m looking for is ways that they can actually TEST probabilities. The Monty Hall problem is surprising, but they can easily play several times and really see the probability “working.” I wonder if there’s a dice game we could model after the cancer test example.

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Nice lesson. Even nicer insight by student in the back of the room! One thing I’ve always struggled with when doing this example with students is that they need to have the insight that when you have 50 bowls, the host is taking away ALL of the bowls except 2. Why doesn’t the host just take 1 of the remaining bowls away (where you would actually still want to switch)?

This isn’t necessarily helpful when you are trying to teach the basics of the subject, but I’m personally motivated by probability when faced with a situation that is counter to my intuition (like the Montey Hall Problem). Another example I really like is the following game…

You choose a sequence of heads and tails, say THH. Your opponent does the same, say HHT. We then play the following game where we flip a coin until one of our sequences appears. So in the game HTHTTHH, player 1 would win after the 7th round. So…I’ll go ahead and be player 1. Anyone want to play?

It was a great insight to think of 50 bowls, but actually that student was about to argue AGAINST switching having better odds (which was great!) I interrupted him and hijacked it at the end.

I had the same problem as you, initially – why does the host open all the doors instead of just one of the doors!? The answer is that even if the host DID just open one of the doors, the odds would still be better if you switched. Crazy. I love that getting information about a seemingly independent choice can affect the odds like this

I really like the mysteries of coin flipping strings. In fact, I assigned some coin pattern investigations as an honors assignment and put together a quick app at http://flippingquarters.appspot.com/ to help them with specific odds (put in a big number like 100000, or if you have Chrome, 10000000). It only looks for HTH and HTT, but there are interesting comparisons between HH and HT as well (easier for students to grapple with).

This year I used the game Settlers of Catan as part of my intro to probability. I used an online App for the game so that teaching the kids the rules of the game took a bit less time.

Anyway, initially I tell the students nothing about the odds of each roll coming up, and I let the students play a game. Even the least imaginative mathematician by the end of the game realizes that their settlement built on a 2 or 12 wasn’t a very good idea as they watch the person who built on 6 or 8 rake in the resources.

Here’s a good picture (and blog post) about Settlers of Catan: http://rulesoflunch.wordpress.com/2008/01/09/nerd-alert/

You can see from the picture that numbers from 2 to 12 are placed at the intersection of a hexagonal tiling pattern. Each turn two 6 sided dice are rolled, and totaled to determine which resource is produced.

Great simulator here:

http://www.shodor.org/interactivate/activities/AdvancedMontyHall/

Here’s what I love about this: I have been railing for ages against using the Monty Hall problem because it is such a counterintuitive “gotcha” situation. It’s so hard to understand, my usual approach would just leave kids flummoxed, seeing Yet Another Unfathomable Mystery. But the fifty bowls are flat-out gorgeous and make the reasoning accessible. So bravo. I’m converted.

I’ve always found visualizing helps students see “the bigger picture”, which seems obvious to say but is easily left out when dealing with these kinds of easily confusing questions, which is why the chalk drawing is so great. For the Cancer question, I would say alright, let’s look at 1000 people. 8 of them have cancer, and 90 percent of them will test positive, so 7. But, for the 992 others, 7 percent of them also test positive, so 64. Altogether 71 people test positive, but how many really had cancer?

“I thought I would like teaching probability, but I don’t, because it claims to be so practical and is actually so impractical.”

What is often impractical is the excess of calculations that go into probability problems. Now, to understand problems like Monty Hall or the physician’s dilemma (the disease problem that Josh G and Drew mention), one approach would involve conditional probability equations and lots of ugly notation. Another would be to diagram it graphically, which often turns out to be the more elegant approach as well.

In “real life,” what is far more important that detailed calculations is the ability to ballpark probabilities. Top poker players, for example, know that their chance of hitting a flush with two cards to come (with 4 to the flush on the flop) is just under 35%. In almost any actual scenario though, ballparking it to 1 in 3 is plenty precise and accurate for pot odd calculations. When I buy ski pass insurance for my season pass, I only have to ask myself whether my chances of a season-ending injury are greater than 5% (which they certainly are). I think an important question to ask, along with “what is the answer,” is one of relative magnitudes and what an acceptable margin of error is. It is at this level that probability is generally applied (or misapplied) in everyday life. I’d like to see more a more conceptual approach (backed, of course by some calculations). Fundamentally, we are not training actuaries, but rather people who have to make frequent decision with incomplete information.

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Someone should ask “When do you decide to show an empty bowl and make the offer to switch?” If you only make that offer when the contestant has chosen correctly ( … ) .

However, if you announce in advance that you’ll make the offer as soon as the contestant chooses a bowl, you get a different set of odds. It’s the latter set of odds that has been presented here.

One reason people find Monty Hall confusing is this: after they look at the bowls (or the grass), they “get it”, but then they think: “Wait – if I just walked in after the reveal and saw two doors, the odds would be 50%! So how can it be 2/3, even though I just realized it must be 2/3!” And the mind is boggled by this seeming contradiction. But this moment of confusion is the door to the key insight about probabilities: they do not depend solely on the situation, they depend on what you know. A person walking in and knowing only that the prize is between one of two doors would say the probability is 50%. A person who saw the reveal would say the probability is 2/3. They are both right.

The probability is not subjective. Anyone seeing only two possibilities must conclude the probability is 50%. Anyone who saw the whole reveal must conclude the probability is 2/3. But probability depends on the situation and on what you know about it.

More obvious example. I flip a fair coin and hide the answer from you but look at it myself and see tails. For you the odds are 50% that it is tails. For me the odds are 100% that it is tails.

Once people get that there are different answers for what the probability is for a given situation, depending on what’s known, a lot of the vertigo goes away, and a deep lesson about the meaning of probability has been learned.

Recommended reading on this:

E.T. Jaynes.

Thanks for this perspective, Dan. There’s a lot to say here about the nature of probability – different amounts of knowledge lead to different estimates of probability. In the end, something happens (unless you never open the box), and that’s 100% true.

I love talking about these things with kids, but I try to be careful not to explode their heads with my own ideas. If their heads are going to explode, I want them to do it to themselves

“If their heads are going to explode, I want them to do it to themselves.” Amen, brother.

It’s an excellent insight about where the confusion comes from, but I disagree with the conclusion.

After the empty bowl is revealed, probability that the prize is in the other bowl is 2/3, independent of any person. And if anyone wants to estimate the probability, s/he should better make sure that the odds of every event happening are equal. In your coin example, after the flip, the odds for the events are not equal – they’re 0 for heads and 1 for tails, it’s just that I don’t know it. Doesn’t change the probability, although it might change my estimation if I’m not careful.

It’s a good point to discuss the condition that each and every event has to have an equal chance of happening with the students.

Oh well, mixed it up myself – the probability that the prize is in a specific bowl is either 0 or 1 immediately after the prize is placed there. According to some interpretations. Just learned that the term ‘probability’ has several interpretations. Good!

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I usually do the whole game with a deck of cards. At first I have the ace of spades and two other nondescript cards. I mix them around, make a big show of looking at all three of them, ask them to pick, reveal one, and offer the opportunity to switch.

Then it’s really easy to switch to 52 bowls, and with the ostentatious looking and revealing, it’s really obvious to most of them that they should be betting on me holding the ace of spades as my one unrevealed card.

What has often surprised me is how hard a time I have getting some of them to see that the 3-card game has the same basic structure as the 52-card game.

Ah, a deck of cards is a great idea. I like being able to throw candy out to the class, but I’m sure there’s a way to work that in too, and cards seem like they’re so much easier to reset (I had to have the class close their eyes while I reloaded the bowls, heheh).

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