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4-Block Stalagmites is part of the ProbLab middle-school curricular material for learning probability. Related materials are a random generator called a "marbles scooper" and a sample space called a "combinations tower". In classroom activities, students working with 4-Block Stalagmites will have interacted with the marbles box and built its sample space and thus would have inferred expectations as to outcome distributions in hypothetical experiments with the marbles box.
4-Block Stalagmites is designed to enable users to experience insights into the binomial phenomenon and in particular to witness and understand the emergence of experimental outcome distributions that, by and large, are both consistent and in accord with proportions in the sample space. The model includes an interactive simulation of a binomial experiment with a sample size of four, which is comparable to an experiment of tossing four coins over and over, only that the four "coins" can land on green or blue, not heads or tails, and each "coin" has a fixed position in a 2-by-2 table that we call a 4-Block, unlike a set of four coins that has not inherent structure and lands "all over the place" on the table.
A unique feature of the model is that the outcome distribution is composed of the actual experimental samples themselves that are stacked one above the other in their corresponding columns. This is different from classical histograms that do not record which specific samples were taken but only their aggregate properties. For example, the particular 4-Blocks sampled appear in the distribution, rather than as just a record of the fact that a 4-Block with 3 green and 1 blue squares (in any order) were sampled.
The outcome distribution is in the form of "stalagmites" of stacked samples that have "dripped" down into their correct column. This creates a picto-graph histogram that grows bottom-up like a stalagmite. When the probability in the model is set at 0.5, this stalagmite will grow to 1:4:6:4:1 proportions. For other p values, the stalagmite will be tailed.
There are four unique 4-blocks that each has exactly one green square, but there are six unique 4-blocks with exactly two green squares each. So, for a p value of .5 (when independent squares are equally likely to be green or blue), it is 1.5 times more likely to draw a two-green 4-block than a one-green 4-block (the ratio value of 6 to 4 is 1.5). This is worthy of attention, because students often need help in understanding how permutations are relevant to combinatorial analysis and, moreover, how combinatorial analysis is relevant to predicting the outcome distribution. When the sorting and coloring effects are activated as the simulation is running, the visual effect of the growing stalagmite is as though it is "stretching" the sample space of 16 elemental events into an outcome distribution of about 160 samples. Within columns we always expect all the elemental events to occur as frequently. However, this is true between columns only for a p value of .5. Note that we can change the p value and thus affect the overall shape of the stalagmites. For example, for the p value of .6, the two-green 4 Blocks will not occur as often as they would for a p value of .5, but the three-green 4 Blocks will occur more often than for a p value of .5. Due to the specifics of this change, the two-green and three-green columns are anticipated to be equally tall.
4-Blocks are randomly generated by asking each square to choose a color with a 'probability-to-be-target-color' chance of being green. Each 4-Block sample "drips" down one of the five columns in accord to its number of green squares. The stalagmite distribution can be sorted by type, even as it grows. There are 16 unique outcomes, so sorting the experimental outcomes by type results in 16 groups. These 16 groups are typically of uneven size, even for the p value of .5, but most often their sizes revolve around the average. For example for 160 samples taken, most groups will contain roughly 10 outcomes. You can "paint" these groups to enhance their visual groupiness.
Buttons: SETUP-initializes the View, essentially "emptying" the columns, and resets the variables and monitors. GO ONCE-generates a single 4-Block and sends it down its respective chute, whereas GO does so forever until one of the columns reaches the top of the display. GO-ORG-begins a run in which the samples sort themselves by type (see SORT OUTCOMES, below) ORGANIZE -rearranges outcomes within each column so that identical 4-blocks are grouped; DISORGANIZE -undoes this rearrangement. PAINT-colors outcomes by type so that identical 4-blocks appear of uniform color (the colors themselves are arbitrary-there is no inherent meaning or scaling); UNPAINT-returns the 4-blocks to their original appearance.
Switches: KEEP-REPEATS?-when Off, repeated outcomes are discarded from the Stalagmite. For example, say the simulation has already generated a 4-block with a single green square in the top-left corner. Any time later in the run, if the simulation generates another identical 4-block, it will descend the column and then disappear the moment it hits the stalagmite. But a 4-block with a single green square in the bottom-left corner would be kept, if it had not been generated. When On, repetitions are kept (as in standard outcome distributions). STOP-AT-ALL-FOUND?-when On, the run will end as soon as all 16 unique outcomes of the sample space have been randomly sampled. When Off, the run will continue until one of the columns reaches the top of the display. MAGNIFY?-when 'On,' a blown-up version of newly created 4-blocks is displayed to the side of the column. This helps, because the samples themselves are small and move fast. When Off, no blown-up sample is displayed.
Slider: PROBABILITY-TO-BE-TARGET-COLOR-determines the chance that each independent square in a 4-block will be green. For example, a value of 50 (50% or .5) means that each square has an equal chance of being green or blue, whereas a value of 80 means that each square has a 80% chance of being green and 20% chance of being blue. Monitors: EVENTS FOUND-keeps track of how many of the 16 possible 4-block outcomes have been randomly sampled.
Plot: EVENTS BY NUMBER OF OUTCOMES-shows how the sixteen elemental events are distributed by the number of outcomes sampled for each. When the first sample is taken, that event would be a '1' whereas all the other fifteen events are still at zero.
Setup the model in its default settings (with the 'probability' slider set to the value of 0.5 and the 'magnify?' switch set 'On'), slow down the model, using the speed slider above the View, and press 'Go'. See how a random 4-block sample is generated at the top of the View, just to the left of the stalagmite columns. Count up the number of green squares in this 4-block and see that the 4-block descends down a column bearing the corresponding numeral at the bottom. For example, if there are exactly two green squares in the random 4-block, it will go down the column with a "2" at the base.
Keep running the model slowly. See how samples are stacked on top of each other in the columns. Look closely at these samples and see if you can locate repeated outcomes, for example see if the 4-block with exactly two green squares in a particular diagonal formation occurred at least twice.
Set KEEP-REPEATS? to 'On' and STOP-AT-ALL-FOUND? to 'Off'. Press 'Go.' The columns will fill up until one of them hits the top, causing the run to stop. Compare the heights of the columns. What might you say about the relationship between these heights? Repeat this experiment and see whether any general pattern recurs.
Press SETUP then GO and wait until the run ends. Now press ORGANIZE. What happened? Press DISORGANIZE and then ORGANIZE, and watch the effect on the outcomes in the columns. Now press PAINT under each of the ORGANIZE and ORGANIZE conditions. When the outcomes are both organized and painted, what can you say about the relation among the sizes of the colored groups? That is, over repeated trials, is there any pattern in the relative sizes of these groups, or is it completely arbitrary?
Set KEEP-REPEATS? to 'Off' and STOP-AT-ALL-FOUND? to On. When you press GO the model will keep running until it has randomly sampled all of the unique outcomes in the sample space. How many samples, on average, are required in order to fill the entire sample space? Does this number change according to the settings of the probability? For example, if the probability is set at 80%, does it take as many trials to fill the sample space as compared to a setting of 50%? If not, why not? How about the extreme cases of 0% or 100%?
Add monitors and/or graphs to explore aspects of the experiments that are difficult to see in the current version. For instance:
How many trials does it take for the experiment to produce an all-green 4-block? How is this dependent on the various settings?
Are there more samples with an even number of green squares as compared to those with an odd number of green squares?
How symmetrical is the set of stalagmites? How would you define "symmetry?" How would you quantify and display its changes over time?
Some of the other ProbLab (curricular) models, including SAMPLER-a HubNet Participatory Simulation-feature related visuals and activities. In Stochastic Patchwork and especially in Sample Stalagmite you will see larger blocks, such as an arrays of green and blue squares. In the Stochastic Patchwork model and especially in 9-Blocks model, we see frequency distribution histograms. These histograms compare in interesting ways with the shape of the stalagmites in this model.
Thanks to Dor Abrahamson for the design and of this model as well as the implementation of the original model. Thanks to Josh Unterman for implementing the advanced procedures. This model is a part of the ProbLab Curriculum, originally under development at Northwestern's Center for Connected Learning and Computer-Based Modeling and now also at the Embodied Design Research Laboratory at UC Berkeley. For more information about ProbLab, please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.
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Copyright 2006 Uri Wilensky.
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