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If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web 
Equidistant Probability connects between probability and geometry. You select two or more squares, and the model searches randomly for squares that are equally distant from the squares you selected. To do this random search, creatures pop out of each one of your selected squares and simultaneously step forward one step in some random direction. If they all land in the same square, that's a hit. Can you guess how often this will happen?
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 user sets up an experiment by selecting some squares on the view. When the model runs, from each of these selected squares emerges a creature, headed in a random direction, and then all these creatures step forward at the same moment. Because there are exactly as many creatures as there are selected squares, and because they all step forward at the same moment, if all the creatures arrive at the same square at the same moment, then that square is "equidistant" (equally far away) from all the selected squares. In that case, the square where they all landed becomes green and keeps a count of how often, out of all their "attempts," the creatures met there. Note that the creatures don't need to land on the exact same spot in the view (the same pixel)  it's enough that they land in the same square.
There is a procedure in this model, "Epicenter," that helps think about how often a single creature might land in its neighboring squares. If we knew that, we might be able to figure out how often two or more creatures might land on the same square at the same time. For instance, if one creature has a 1in2 chance of landing on some square, and another creature has a 1in3 chance of landing on that same square, then there's a 1in6 chance that they will land on that square at the same moment. In this procedure, 10,000 creatures pop out of the middle square and take a step forward in some random direction. The other eight squares count up how many creatures landed on them, and these values are shown either as totals or as percentages.
Here is a quick way to get familiar with the model. Press "PRESET1," slow down the model (using the slider on the topleft corner of the view), make sure SINGLESUCCESS? is at "Off," then press FIND EQUIDISTANT POINT. Watch how two creatures first emerge, each from its red square, and then step one step forward in some random direction. Once you understand this, speed up the model gradually. Quite soon, the creatures will land on the same square at the same time, that square will become green, and it will display a percentage, for instance 5.00% (that is, 1/20). That would mean that it took 20 attempts to find that square. Also, the plot ATTEMPTS UNTIL SUCCESS will show a histogram bar at "20," and the average that is, simply, "20" at this time. Also, the monitor below the plot, MEAN ATTEMPTS TO SUCCESS, will show "20." Now press FIND EQUIDISTANT POINT again. It could be that a different square will be found this time. The plot and monitor will update, again. If you switch the SINGLESUCCESS switch to "Off" and press FIND EQUIDISTANT POINT, the model will keep searching and finding. Watch the monitor SAMPLES TAKEN UP TO NOW to see how many attempts have been taken towards completing a sample (the default setting of SAMPLESIZE is 1,000). Once a sample has been completed, the plot #SUCCESSES PER SAMPLE will show how many successful attempts there were in that sample.
Buttons: SETUP  initialize variables SELECT SQUARE  once you press this, you can click on the view to choose squares that become red. If you click again on a red square, it will be unselected. When you are through with selecting, unpress the button so as not to make further selections by mistake while the model is running. FIND EQUIDISTANT POINT  sets off the search procedures. From every red square, dartlike creatures pop up, each headed at some random direction, and then they step forward. PRESET 1 through 6  each 'Preset' button sets the model up with a preselected configuration of red squares. EPICENTER  activates a procedure in which 10,000 turtles emerge from the center square at random orientations, and all step forward a single step. Pressing this causes you to lose all data from an experiment that you may have been running.
Switches: SINGLESUCCESS  when "On", the procedure FIND EQUIDISTANT POINT will stop the moment a square has been found that is equally distant from the red squares. When "Off", the procedure will continue again and again, until you unpress FIND EQUIDISTANT POINT.
Monitors: #SUCCESSES  shows how many successes you have had in this experiment, that is, how many times all the creatures landed at the same time in one and the same square. #ATTEMPTS  shows how many attempts you have had in this experiment, that is, how many times all the creatures have "tried" to land at the same time in one and the same square. FREQUENCY  how many successes there have been out of the total number of attempts, expressed as a percentage. This can never be larger than 100 because there cannot be more successes than there are attempts. SAMPLES TAKEN UP TO NOW  shows how many samples have been completed, for instance of 1,000 attempts each, and how many attempts have been taken in the current sample.
Plots: ATTEMPTS UNTIL SUCCESS  a histogram that updates each time an equidistant square is found, to show how many attempts were needed to find that square.
SUCCESSES PER SAMPLE  a histogram that updates each time a sample has been completed, for instance of 1,000 attempts, to show how many successes there were in that sample.
Choice: DISPLAY RESULTS (relates to the 'Epicenter' button) 'bytotal'  squares show the number of creatures that landed on them. 'by%'  squares show the percentage of creatures that landed on them out of all landings.
The creatures' steps are equal in length to the side of a square. If a creature has just emerged in the middle of a square and is heading at 90 degrees, then the step forward will land it at exactly the middle of the square to its right. But if it's headed at 45 degrees, then the step forward will put it in the square that is diagonally neighboring it on the top right, short of the middle of that square.
The values of 'Frequency' and 'Mean #Attempts to Success' are reciprocal:  Frequency is what you get when you divide #Successes by #Attempts  Mean #Attempts to Success is what you get by dividing #Attempts by #Successes. So if you multiply these values, you get 1 (or 100, if you ignore the "%" sign).
For Presets 1 and 2, the creatures will find more than a single square that is equally distant from the red squares. But for Presents 3, 4, 5, and 6, they will only find one such square.
When you press EPICENTER with the choice set at "by%," the percentages you get are such that the North/West/South/East squares each has roughly 1/6 of the creatures (16.7%), and the cornering patches each has 1/12 of the creature (8.33%).
Set the model with Preset1 and run the model with SINGLESUCCESS? set to "Off." Note that the middle square will register four times as many successes as compared to the centerleft square and as compared to the centerright square. Try to come up with a logical explanation for this.
Set the model with Preset3 and run the model with SINGLESUCCESS? set to "Off." Track the values you get in the monitor MEAN #ATTEMPTS TO SUCCESS. Try to find a different selection of squares that gives the same value. What does this selection have in common with the Preset3 selection? Repeat this for the preset conditions 4, 5, and 6.
A very big challenge: The experiments in this model are all based on random choices. After enough attempts, the results begin to stabilize around certain values. Can you determine what these numbers will be before you run an experiment? For instance, can you anticipate the value of MEAN #ATTEMPTS TO SUCCESS on the basis of the configuration of squares that you have selected? To do this, you could think about the information you get from the "Epicenter" procedure. Also, you may want to analyze this problem using pencil and paper.
Add a plot for the cumulative ratio of '#Successes' to '#Attempts.' The plot should update at every attempt.
Edit the size of the view so as to include more squares. You will be able to create configurations that you could not create before.
For the "Epicenter" procedure, increase the number of squares in the view. Add interface widgets and code that allow for broader experimentation, such as variety in the number of steps the creatures take and the size of these steps.
The three different ways of looking at the data in this model are the same as in Prob Graphs Basic: cumulative ratio, attempts until success ("waiting time"), and sampling.
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/.
Additional reading: Abrahamson, D. & Wilensky, U. (2003). The quest of the bell curve: A constructionist approach to learning statistics through designing computerbased probability experiments. Proceedings of the Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italy, Feb. 28  March 3, 2003. Available for download at http://ccl.northwestern.edu/papers/2003/AbrahamsonWilenskyCERME3.pdf
If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.
For the model itself:
Please cite the NetLogo software as:
Copyright 2004 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.
This model was created as part of the projects: PARTICIPATORY SIMULATIONS: NETWORKBASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs)  grant numbers REC #9814682 and REC0126227.
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