We’re starting to learn about probability. Surely one of the quintessential settings is rolling two dice and adding. I’ll try to walk that back another time and rationalize why I include it, but for now, I want students to be able to explain why seven is more likely than ten. I want them to have that archetypal diagram in their heads.

But starting with the theoretical approach won’t go very well. Furthermore, with my commitment to data and using randomization for inference, an empirical approach seems to make more sense and be more coherent. So that’s what I’m trying.

The key lesson for me for this report—related to “trust the data”—is that actual data, with the right technology, can illuminate the important concepts, such as *independence*. This makes me ask how much theoretical probability we need, if any.

### What Happened in Class

To do the Dice Sonata (previous post), I had given each student two dice: a red one and another one. They rolled them 50 times, recording each result twice: once to do the sonata, so they could make the graph of actual results by hand, and also on the computer in a Fathom Survey so we could easily assemble the results for the entire class.

If you haven’t used Fathom Surveys, you can think of it as a Google form that you can later drag directly into Fathom. The key thing here is that they recorded the red die and the other die *separately*. When we were done, we had 838 pairs.

This was Thursday, the second class of the semester. After students discussed the homework, and saw that their sets of 50 rolls didn’t produce graphs with their predicted shapes, we went to the computers to see if things looked any different with more data. To make the relevant graph, students had to make a new attribute (= variable = column ) to add the two values—which they already knew how to do. Here is the bottom of the table and the graph:

One could stop here. But Fathom lets us look more deeply using its “synchronous selection” feature (stolen lovingly from ActivStats): what you select in any view is selected in all views.

Continue reading An Empirical Approach to Dice Probability