Embodied Design Research Laboratory

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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/ and http://edrl.berkeley.edu/design.shtml.

WHAT IS IT?

4-Blocks simulates an empirical probability experiment in which the randomness generator is a compound of 4 squares that each can independently be either green or blue. The model helps us conceptualize relations between theoretical and empirical aspects of the binomial function: combinatorial analysis (what we can get) and experimentation (what we actually get).

HOW IT WORKS

The model operates all four squares simultaneously, just like flipping four coins at once. At every Go, each square "flips a coin" to decide whether it should be green or blue at a probability you have set. The number of green squares in the block is counted up and added to a list that is plotted as a histogram. Over many runs, the histogram begins to Òtake shape.Ó

PEDAGOGICAL NOTE

There are five possible compound events (combinations): no-green, one-green, two-green, three-green, or four-green. Within these events, there is a total of 16 unique outcomes (permutations) comprising the sample space: 1 no-green, 4 one-green, 6 two-green, 4 three-green, and 1 four-green. These numbers, 1-4-6-4-1, are the n-choose-k coefficients of the binomial function, where n is 4 and ÔgreenÕ is the favorable event. A histogram tracks the accumulation of these compound outcomes. When the probability of getting green is at .5, the emergent empirical distribution will dynamically converge on a 1:4:6:4:1 distribution, and when we offset the p value of .5, the distribution is accordingly offset. A "stratification" feature allows us to monitor within-column accumulation corresponding with the 16 unique permutations.

HOW TO USE IT

Buttons:
'Setup; - initializes the variables and erases the plot.
Go Once - activates the procedures just once. So you will get a single 4-block in the View and the corresponding column in the histogram will rise by one Ònotch.Ó
Go - activates the procedures repeatedly until you press it again.

Slider:
Probability-to-be-Target-Color: Set the chance for each of the independent squares to be green.

Switches:
One-by-One-Choices? Ð A feature to help users see the 4-Block as a set of four independent randomness generators (like four flipping coins). When On, each square will settle on its color at a different moment. Also, when Go is pressed, there will be a pause between 4-blocks. YouÕll want to set this switch to Off once the nature of the experiment is clear.
Stratified? Ð When On, the histogram columns are each partitioned to reflect accumulation by unique outcome. As more samples are taken, you will see the 1-4-6-4-1 structure emerge; the within-column blocks will equalize, but the between-column blocks will be equal only for a probability of 50%.

Monitors:
Total Trials - shows how many random 4-blocks have been generated in the current experiment.

THINGS TO NOTICE

Using the default model settings, press Go Once. See how each of the four squares takes time to decide whether it is green or blue. This is like four coins that are spinning on the table until they each settle either on Head or Tail.

THINGS TO TRY

Run the model with the probability-to-be-target-color set at 50%. As the model runs, the histogram grows. Pretty soon, the central column grows taller than other columns. This demonstrates that there is a higher chance of getting 4-blocks that have exactly two green squares as compared to 4-blocks that have either zero, one, three, or four green squares. But there is more structure to interpret. The lowest chance is to get a 4-block with no green squares or with four green squares. The chance of getting a 4-block with either exactly one green or three green squares is in between.

Now change the value of the probability slider, press Setup, and run again. What do you see?

Set the Stratified? switch to On, the model speed to slow, and the probability value to 50% Run the model. See in the histogram how new blocks pop up. Eventually, the number of blocks in the columns will be 1, 4, 6, 4, and 1, respectively.

NETLOGO FEATURES

Auto-Plot: Look at the histogram as it grows. What happens when it reaches the top? More and more combinations are coming but there is no room to count them. Instead of leaping out of the boxÉ, the number at the top-left corner of the plot -- the value of the y-axis Ð updates, and the histogram is redrawn to fit.
Stratification: The partitioning of the histogram columns by unique outcome was creating specially for this model. See the procedures to learn how this was accomplished. Essentially, a nested structure was used to create within each column as many sub-columns as necessary. For example, the one-green columns (second from the left) is comprised of four overlapping columns.

EXTENDING THE MODEL

You may want to monitor different aspects of the probabilistic experiment, to answer such questions as:
- how often do we get the same combinations twice one after the other?
- is there particular permutations you like? You could add code to see how long it takes the model to find this permutation.
- what is the dynamic ratio between the number of one-green and three-green outcomes?

CREDITS AND REFERENCES

This model is a part of the ProbLab curriculum. The ProbLab Curriculum is currently under development at UC BerkeleyÕs Embodied Design Research Laboratory (http://edrl.berkeley.edu/). For more information about the ProbLab Curriculum please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.

To refer to this model in academic publications, please use: Abrahamson, D. & Wilensky, U. (2004). NetLogo 4-Blocks model. http://ccl.northwestern.edu/netlogo/models/9-Blocks. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.