## Dice

Dice 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 "Dice," a virtual laboratory for studying fundamental probabilistic aspects of a classic pair of dice, each with 6 sides. The model simulates randomly rolling dice and uses statistical analytic tools to make sense of data accumulated over numerous rolls.

For more details, please see NetLogo library. Note that this model is still "unverified" and is yet to undergo our rigorous checkout procedure.

CM ProbLab: Dice
Don't see nothin'?

Gist

Press SETUP, then press PICK DICE. By clicking on the green squares, you set a pair, for instance [3; 4]. Press SEARCH to begin the experiment, in which the computer generates random dice face. If you set the SINGLE-SUCCESS? switch to 'On,' the experiment will stop the moment the combination you had created is discovered. If this switch is set to 'Off,' the experiment will keep running as many times as you have set the values of the SAMPLE-SIZE and TOTAL-SAMPLES sliders. In the plot window, one or two histograms start stacking up, showing how many times the model has discovered your pair in its original order ("Combination") and how many times it has discovered your pair in any order ("Permutations"). Are there always more permutations than combinations?

Questions to Ponder

• Run an experiment with a sample size of 20 and then run it with the same settings but with a sample size of 100 or more. In each case, look at the distribution of the SUCCESSES-PER-SAMPLE DISTRIBUTIONN. See how the experiment with the small sample resulted in half-a-bell curve, whereas the experiment with the larger sample results in a whole-bell curve. Why is this so?

• Pressing HIDE/REVEAL after you create a combination allows you to setup an experiment for a friend to run. Your friend will not know what the combination is and will have to analyze the graphs and monitors to make an informed guess. You may find that some combinations are harder to guess than others. Why is this so? For instance, compare the case of the combination [1; 1] and [3; 4]. Is there any good way to figure out if we are dealing with a double or not? This question is also related to the following thing to try.

• For certain dice you pick, if you run the search under the "Both" option of the ANALYSIS-TYPE choice, you will see only a single histogram in the SUCCESSES-PER-SAMPLE DISTRIBUTION plot. Try to pick dice that produce a single histogram, then try to find others. What do these dice pairs have in common? Why do you think you observe only a single histogram? Where is the other histogram? How do the monitors behave when you have a single histogram?

• When the Combination and Permutations histograms do not overlap, we can speak of the distance between their means along the x-axis. Which element in the model can affect this distance between them? For instance, what should you do in order to get a bigger distance between these histograms? What makes for narrow histograms? Are they really narrower, or is it just because the maximum x-axis value is greater and so the histograms are "crowded?"

• Set the SAMPLE-SIZE at 360 and TOTAL-SAMPLES at its maximum value. Pick the dice [3; 4], and run the experiment. You will get a mean of about 10 for the Combination condition (in which order matters, so only [3; 4] is considered a favored event), and you will get a mean of about 20 for the Permutations condition (where the order does not matter, so both [3; 4] and [4; 3] are considered favored events). Why 10 and 20? There are 6*6=36 different dice pairs when we care for the order: [1; 1] [1; 2] [1; 3] [1; 4] [1; 5] ... [6; 4] [6; 5] [6; 6]. So samples of 36 rolls have on average a single occurrence of [3; 4] and a single occurrence of [4; 3]. Thus, samples of 360 have 10 times that: 10 occurrences of [3; 4] and 10 of [4; 3], on average.

[last updated January 28, 2005]