Download Dice Stalagmite.nlogo (30 KB)
Dice Stalagmite is authored in the NetLogo modeling-and-simulation environment. The model is part of ProbLab, a curricular unit designed to enrich student understanding of the domain. The online unit package will include a suite of models, student worksheets, and a teacher guide. Below is an applet of Partition Permutation Distribution. You can interact with this model by pressing Setup and Go. You may want to slow down the model in order to get to know it -- use the ADJUST SPEED slider that is on the top-left corner of the graphics window. For more details, please see the model itself in the NetLogo library. Note that this model is still under development and is yet to undergo our rigorous checkout procedure.
CM ProbLab: Dice Stalagmite -- Two are better shaped than one
Don't see nothin'?
Dice Stalagmite is a model for thinking about independent and dependent random events. In this model, outcomes from two independent events -- the rolling of "Die A" and "Die B" -- are represented in two different ways. On the left you see these outcome pairs plotted as individual events -- this representation ignores the fact that the two dice were rolled as a pair and just stacks each die outcome in its respective column within the picture bar chart. On the right, you see the same two dice accumulating as pairs. Note that the bar chart on the left has only six columns: 1, 2, ...6 -- one for each possible die face. The chart on the right has 11 columns -- 2, 3,...12 -- because it groups outcomes by total, and there are eleven possible totals (the columns are broader, because they must accommodate two dice).
Below is the combinatorial space of all possible pairs of dice grouped in eleven columns according to their total (2, 3,..., 12). (You can download a Word file of this distribution here).
# Different combinations and arrangements
3 4 5 6 7 8 9 10 11 12
Sum of two dice
As in other ProbLab activities, we care for exploring relations between the anticipated frequency distribution, which we determine through combinatorial analysis, and the outcome distribution we receive in computer-based simulations of probability experiments. To facilitate the exploration of the relationship between such theoretical and empirical work, we build tools that bridge between them. These bridging tools have characteristics of both the theoretical and empirical work. Specifically, we structure our combinatorial spaces in formats that resemble outcome distributions, and structure our experiments so as to sustain the raw data (not just graphs representing the data).
Questions to Ponder
We have also created a Participatory Simulation of this activity, which you can download along with its client.
Note the shape of the outcomes in the right-hand bar chart. In a classroom, discussion should focus on the relation between the theoretical and empirical distribution. Why is it that they are similar? What does it mean? Specifically, if each event is random and independent of previous events, why are we getting a shape that doesn't feel random (we keep getting the same shape)? How can this randomness and determinism coexist? The bar chart on the left hones this discussion, because, from run to run, it is basically a "flat" distribution -- for instance, you can never predict with certainty which die column will be first to reach the top.