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.
50 bowls. One of them has candy in 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!”
Does this bowl have the candy?
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.
The original choice and one other bowl remain. What are the odds that the candy is in the other bowl?
“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?
, the “y-coordinate” column is 
, and the “ratio” column is 
, but the students don’t need those names yet, so I didn’t introduce them. I wanted to meet the students at their own fluency level, and they are perfectly comfortable with x- and y-coordinates, protractors, angles, and ratios (this is a precalculus class).
instead of making a new cosine function for a circle of radius two.