I've been thinking a LOT about reasoning from definitions lately. I think the definition of sample space is the number of *equally likely* outcomes. From that perspective, I think it can be logically argued that the 21 possible rolls of a pair of dice can't be a sample space, because rolling a sum of 3 (1,2 or 2,1) is more likely than rolling a sum of 2. Although maybe the whole point of students' arguments is that they think rolling a 2 and rolling a 3 are equally likely?

My counter-thought-experiment to that is, if you are going to bet on rolling a 3, how many outcomes will make you happy for the first roll? How about for the second roll? What changes if you bet on rolling a 2? Which feels like a safer bet? Why?

I would have done exactly what you did in this instance (at least I like to think I'd have been that wise), and I think that there are many more opportunities for students to encounter and problematize this idea. Better to leave it as a question than do some telling by which you think you've cleared it up and never return to it again! They'll learn this when they encounter the right pile of ideas at the right moment when they have the right questions in their head!

But since I was thinking of definitions, I thought I'd share what came to mind for me about using definitions to make a mathematical claim everything else (like common sense) doesn't clear things up.

Wow, your students sound like great arguers and justifiers. I remember having this same conversation as a senior in college with a professor of probability and feeling less prepared to justify my ideas than your students sound. I certainly never thought of the "how many outcomes for the first roll? the second?" until I was teaching this myself for the 3rd time. I'm curious how you've seen their justification skills get better over the course of the year and what kinds of things you've done to work on that?

I also think it's important to note that this might not actually be a "misconception" if they're asking questions like "is rolling two identical dice at the same time comparable to rolling two single die at separate times"? That means they're getting pretty deep into what it means to think about probability.

I liked Joshua's question about "just how far off are our two different theoretical models from the experimental model?" It seems like the one thing Team 21 hasn't fully convinced me they're accountable too is the discrepancies between the theoretical and experimental models. I'm not convinced it's small enough to chalk up to chance (what a cool way to accidentally launch an investigation into hypothesis testing!)

One thing that's going on in this story that feels extra neat to me is that you've created a situation in which this is a community concern. It's not just a matter of you thinking about one student and their ability to get the right answer to a probability problem individually. It's a situation where a group of mathematicians are working on a problem together and getting different results. I'm curious what you've heard from the students about how that feels? Are they happy to agree to disagree? Or does it feel weird to either group not to be able to come up with a model for dice probability that everyone can accept? I was thinking that if I were home-schooling one member of Team 21 I would feel compelled to keep offering more and more probability experiences to find out what they had/lacked in their mental model that didn't match mine, and see if I could get us on the same page. In a classroom setting where I wasn't one of only two voices I think I would feel less like "I need to keep probing this" and more interested in how this community of mathematicians would cope with discrepancy in their midst. I think.

That said, I love coming up with investigations and so I couldn't help but think of two things I've thought of recently about experimental probabilities. One was in a course I took... Using coin flips to model if a baby would be a boy or a girl and testing an alternative to China's one baby policy which is to let families have babies until they have a boy. Would that increase the population rate? What would it do to the number of girls? I made a prediction then did the experiment, found my prediction was way off, then looked at te theoretical probability to confirm my experimental. I learned a lot... And the issue has been in the news recently b/c of the lack of women in many rural villages leading to whole towns of just bachelors (and also bride stealing and selling).

I also thought of something I overheard recently which was that even though theoretical probability suggests it's more likely you'll get struck by lightning while winning the Nobel prize than win the lottery people do it (win the lottery) every day. So the person was saying that this is the difference between experimental and theoretical probability. That seemed weird to me, like it suggested you shouldn't trust either probability or like they had nothing to do with each other. So it seemed like a neat investigation: if the lottery is so hard to win, why do people win? Is that a problem for theoretical probability?

Another thought I just had was to use a rectangular area to model the "sample space". Create a grid that looks like an empty multiplication with the numerals 1-6 down one edge (marked Blue Die) and also across the top (marked Red Die). Have the kids help you fill in the sums. They will notice that there is a single instance of 2 as a sum, and that there are two instances of a sum of 3 (see my earlier post).

An area model is a good strategy for them to have in their back pocket
to model other probability situations, too.

Bryan,

You said: "I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think." Does that mean you did not explain to the "21 camp" why they were wrong?