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Dice Stalagmite is a model for thinking about the relations between independent and dependent random events. Pairs of dice are rolled, then the dice fall into columns in two bar charts. One of these charts records the dice as two independent outcomes, and the other, as a single compound event (sum) of these two outcomes. Because the columns grow from the bottom up, we call this a "stalagmite."
Different distributions emerge: the independentevent bar chart is flat (equally distributed) whereas the dependentevent bar chart is peaked. (It does not quite approach a normal distribution, because there are only two compound outcomes.)
This model is a part of the ProbLab curriculum. The ProbLab curriculum is currently under development at the CCL. For more information about the ProbLab curriculum please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.
The outcomes from rolling the two dice are represented in two different ways.
On the left, they are plotted as individual events. This representation treats the dice individually, not as pairs. Each die is stacked in its respective column, one through six, in the resulting histogram.
On the right, you see a second histogram with the same dice stacked in pairs according to their sum. There are eleven columns, 2 through 12, since those are the possible sums of two dice.
When the model is run, the right chart never reaches the top before the left chart. (Why?) The left bar chart is "bumped" down by one row so as to leave more room for the bars to grow. This allows for the bar chart on the right to grow further and take on its typical (peaked) shape.
Switches: STOPATTOP?  if 'On', stops the model when the right side of the display bar chart (the dice totals) has reached the top. If 'Off', then both stacks "bump" down one row when a column hits the top. (The plots on either side of the view are always scaled to show all of the data, even if the view is only showing the top portion.)
Buttons: SETUP  prepares the model for running.
GO  runs the model. In a single run of GO, a random pair of dice appears, is copied, and then the copies fall into their stacks. Also, the plots are updated.
Plots: SINGLE DICE  plots the number of occurrences of each dienumber (16). PAIR SUMS  plots the number of occurrences of each dietotal (212). The plots show the same information as the view, except that the plots always show all of the data, while if the STOPATTOP? switch is off, the view only shows the tops of the stacks.
As in other ProbLab activities, here we are interested in exploring relations between the anticipated frequency distribution (the relative probabilities), which we determine through combinatorial analysis, and the outcome distribution we receive in computerbased 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). The "picture bar chart" of the combinatorial space of dicepair totals can be found with the ProbLab materials.
Beside each bar chart  the 'dependent' and the 'independent'  there is a histogram that represents the data correspondingly. Whereas the bar charts stack the outcomes so as to sustain the images of the discrete events (the "raw data" themselves), the histograms grow in continuous columns (without partition lines). Twinning each picture bar chart with its respective histogram may help students both to understand the histograms and to shift from additive interpretation of the columns in the picture bar chart (focusing on differences between heights of columns) to a multiplicative interpretation of the bar chart (focusing on the proportions of the column heights).
In a classroom, students should work with the triangular combinatorial space they created (not the one from the model, but one with all 36 different possible outcomes of a dice pair that are arranged in a bar chart). Discussion should focus on the relation between the theoretical and empirical distribution, that is, between the combinatorial space and the distribution of random outcomes. Why is it that they are similar?
Note the shape of the outcomes in the righthand bar chart. The top is triangular. What does this mean? Specifically, if each event is random and independent, why are we getting a shape that is not random (always the same shape)? How can randomness and determinism coexist like this? 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.
If the model runs long enough and if STOPATTOP? is set to 'Off,' you will notice that some columns in the picture bar chart on the left vanish. That is, you will see a die descending to the bottom of its column and "going below sea level" so it is no longer visible. What happens is that this die's column is now too short to appear in the display. It might grow tall enough later to come back in, or it might not. Meanwhile, the histogram in the plot keeps all of its columns, so you can keep comparing between them.
How many pairs are needed until the dicepair bar chart reaches the top? Is this number constant? How much does it vary?
What is the biggest vertical gap between columns in the singledie bar chart? Does the gap get larger or smaller the more you run the model? Does any particular column win more often than others?
Which column in the dicepair bar chart gets to the top first most often?
Currently, the model sums two dice. An interesting idea would be to extend this model to have a sum of three or more dice. There would be more columns for the different dicetotals. How many? How would this change affect the dicetotal distribution?
Currently the model puts all pairs of dice that sum to the same number in the same column. What would happen if you added additional columns so that different combinations were in different columns, for example, so that 2+5 and 5+2 were considered different? Would this change the shape of the dicetotal distribution?
In this model, the origin (patch 0,0) is placed between the single and pair bar charts rather than in the center, which makes computations simpler and extending the model easier.
Dice Stalagmite uses the same basic metaphor as the ProbLab model 9Block Stalagmite. In that model, a random 9block or 4block is selected from a sample space. Then, the block finds is correct column, according to the number of green squares in the block, and stacks up in that column.
The idea of juxtaposing two or more different representations of the same running data is used in several ProbLab models, such as Prob Graphs Basic or Random Combinations and Permutations.
Dice are also used in the ProbLab model Dice for generating a distribution of random outcomes.
The Galton Box model also features raw data that descend and stack up in columns.
This model is a part of the ProbLab curriculum. The ProbLab curriculum is currently under development at Northwestern's Center for Connected Learning and ComputerBased Modeling. For more information about the ProbLab curriculum please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.
Thanks to Josh Unterman for building the original version of this model. Thanks to Steve Gorodetskiy for his contribution to the design of this model.
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Copyright 2005 Uri Wilensky.
This work is licensed under the Creative Commons AttributionNonCommercialShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/byncsa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.
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