Lab 0: A Game of Plinko (Online Version)
What do we do in this lab?
This lab has four parts:
A. Use a simulation of a Plinko game to collect data in order to review concepts related to statistical
uncertainty
B. Review a model for how one variable in the experimental system relates to another
C. Design and conduct an experiment to assess a model for how the average result changes with the
binary probability
D. Consider what conclusions can be drawn from the experiment and how to write an abstract
summarizing everything altogether
Equipment: PhET Simulation, Spreadsheet software
Key Concepts: Mean, Standard Deviation, Standard Deviation of the Mean, Linear Model, Binomial
Distribution
Safety concerns: None particular to this lab. But for all online courses: make sure to take breaks to avoid
eye strain from viewing your computer screen! Take care.
Part A. Experiments and Uncertainty
A primary goal of empirical science is to compare theories (often written in a mathematical language)
with the real world in order to assess their validity.
In this class we’ll use the phrase “Physical Model” as the label for the theories we’ll be assessing, and
“Experimental System” for the real world system we’ll be comparing to (although this semester this will
often be a simulation thereof). In the process of doing this we’ll encounter the need for various
“Statistical Methods” to allow us to make this comparison.
In this lab we’ll work through a first example of all of it. You will not need to turn in a report, but you
should complete the activity in class and reflect on your results and answers. You’ll be given an instructor
provided report at the end of the week. Comparing your own results and reflections with what you’ll be
provided will help prepare you for writing your own reports in the future.
A1. Open the simulation:
https://phet.colorado.edu/sims/html/plinko-probability/latest/plinko-probability_en.html
Select “Lab”.
You’ll see a Plinko board with some options for dropping balls down it. This will be the “Experimental
System” for this lab, which we’ll try to compare with a mathematical model. Set the rows to 16, leave the
Binary Probability as is.
A2. We may wonder what will happen if we release a ball from the top. Where will it land? Hit the “play”
button to release a ball.
Notice your result is displayed below the board, and that summary statistics are proved on the bottom
right.
A3. A key idea in science is that when we repeat experiments, we get the same result. With this in mind,
release a few more balls. Do you get the same thing each time?
A4. Probably not, but all is not lost! This is because any real experiment involves some uncertaintycollections of unknown and often random influences on the system. Much of empirical science revolves
around understanding uncertainties.
In this case the uncertainty is random and we can handle it using some “Statistical Methods”. Try
releasing more balls until you have at least 10 trials. You’ll note that although you don’t get the same
result for every trial, a clear distribution emerges.
This is a key idea: the result of an experiment is a distribution, not a single number.
A5. Often we’re interested the average behavior of a system, since this eliminates the many inevitable
random influences. We can compute a mean value from our data to try to determine this. In fact, the
simulation does this for you, on the right. We’ll go into more detail in future weeks, but for reference the
formula is:
=
1 + 2 + ⋯
However there’s always some uncertainty in the result related to the width of the distribution. We can
characterize the width with the standard deviation, which is also displayed on the right. Again, we’ll go
into more detail in future weeks, but for reference the formula is:
= √
( 1 − )
2 + ( 2 − )
2 + ⋯ ( − )
2
− 1
Strictly speaking, the uncertainty in the mean value itself can be smaller than the standard deviation.
Often we are justified in using the standard deviation of the mean, and it’s displayed as well. However in
this course we’ll be more conservative and only require you use the standard deviation as the uncertainty
unless otherwise stated. (However you’re welcome to use the standard deviation of the mean if you
choose as long as you justify it. In general this is the case: you must justify the statistical methods you
employ).
A6. We can display the result of our repeated measurements as:
Best Estimate +/- Uncertainty (with units).
For example, perhaps you got:
8.2 +/-2.7 intervals
Notice that although there may have been some rounding here, which is fine, the decimal precision in the
best estimate and uncertainty match. We’ve called the distance units “intervals” since we’re not given an
actual distance scale.
Part B. A Model Relating Variables
Let’s now consider a model which relates the mean value of our result to another variable which we can
control. We usually refer to the latter as an independent variable and the former the dependent variable
(which is what we measure).
B1. We see that we can change the Binary Probability to be more or less than ½ using the right menu.
This will cause a tendency for the ball to go more towards the right or left. How will this affect the mean
value? A simple “Physical Model” we could propose is this linear one:
=
Where we have the average result as the dependent variable, the probability as the independent
variable, and the number of rows a parameter in the model.
Why this model? In some sense it doesn’t matter for our purposes, we can decide just by testing. However
we could justify it as a hypothesis by saying that the ball moves rightward with probability once for
each row, so the mean should be proportional to both. (Or else we could use the binomial distribution, if
you’re familiar with that, but it’s unnecessary).
B2. Let’s collect some data to assess the model. For 5 different choices of run 10 trials each and record
the results in a table like this one:
Part C. Assessing the Model
C1. We will learn more sophisticated techniques for comparing data with theory in future labs. For now,
just examine the results and their uncertainties: does each result match with the prediction of the model?
If so, we can regard the model as accurate, at least within the stated uncertainties. If not the model is
rejected and will need to be modified.
Part D. Conclusions and Abstract Summary
D1. Try writing a brief conclusion summarizing the results you found in this experiment.
D2. Then, write an abstract meant to briefly summarize everything in the lab concisely.
Probability p Mean of 10 trials Standard Deviation Predicted Value
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