The Marbles Scooper: A random generator for probability experiments.
The marble scooper is a device for sampling a fixed number of marbles out of a vessel containing many marbles, such as a box with equal numbers of green and blue marbles. We have built scoopers that sample exact numbers of marbles—a 4-Block marble scooper and a 9-Block marble scooper. Here we willl feature the 4-Block marble scooper that takes samples such as 3 green marbles and 1 blue marble. The marble scooper was used in the Seeing Chance project.
Whereas technically speaking the marbles scooper random generator simulates a hypergeometric experiment, because each sampled marble is not returned to the box before the next one settles in, the ratio of sample size (4 marbles) to the population (hundreds of marbles) make this an approximation of the binomial.
The 4-block is a 2-by-2 grid in which each cell is randomly assigned one of two colors, e.g., green or blue. Thus, the 4-block can take on 16 unique configurations (patterns). These 16 patterns can be grouped by the number of green cells in each, forming groups with 1, 4, 6, 4, and 1 marble(s), corresponding with the sequence of coefficients of the binomial function (a + b)4 (i.e., there is just 1 block with zero green cells, there are 4 unique blocks with exactly one green cell, 6 unique blocks with exactly two green cells, 4 unique blocks with exactly three green cells, and a single block with four green cells). Furthermore, these five groups can be arranged such that they form a "histogram" of vertically-stacked columns of heights 1, 4, 6, 4, and 1 -- the combinations tower.
The Combinations Tower -- distributed sample space of a binomial experiment
The scooper is a unique stochastic object:
(a) Stochastic function decouples object and outcome: Unlike the case of coins or dice, a particular marble-scoop sample is not an inherent physical aspect of the device but is constituted only through an interaction between the device and the "population" of marbles. Decoupling the stochastic object from its outcomes, the scooper may help in conceptualizing the combinatorial space of the 4-marble compound event as a spatial "variable" with color "values."
(b) Topology supporting unique identity of permutations: Scooping with the 4-block is logically and mathematically commensurate with tossing 4 coins—both operate as a set of 4 independent randomness objects that can each take on two different values (green or blue; Heads or Tails). Yet the 4-block is structurally different from a set of 4 coins—it affords a different experience that is designed to support the construction of a combinatorial space as well as an understanding of probability experimentation, as I now explain: With tossed coins, it is difficult to monitor which coin is which, because the coins are completely identical. Therefore, coming from the experience of tossing coins, it may be difficult to initiate a combinatorial analysis in which, say, 'Heads, Heads, Heads, Tails' is designated as distinct from 'Heads, Heads, Tails, Heads.' The 4-block, on the other hand, assigns fixed locations to each of the four cells. The 4-block may thus scaffold student attention to the uniqueness of permutations. Understanding the relevance of permutations to combinatorial analysis is critical for coordinating between the combinatorial space and outcome distribution (because the scooper "knows" all about permutations).
(c) Interaction supporting a temporal–spatial bridging (sequential to simultaneous): With coins, it may not be clear that four consecutive tosses of a single coin are commensurate with simultaneous tosses of four coins. The 4-block is an inherently simultaneous and not sequential randomness device, and so the problem of appreciating that the sequential and the simultaneous are commensurate is a priori eschewed. This is definitely not to say that this learning challenge should be avoided. Rather, the 'Sequential vs. Simultaneous' would be regarded in my proposed framework as a learning axis of the domain of probability that is not expressed as a learning issue in this particular design.
Developing the Marbles Scooper -- from idea to reality.
For publications of research using the marbles scooper, please see the Seeing Chance page.
Credits: Designed and researched by Dor Abrahamson. Thanks to Paulo Blikstein, then at the Center for Connected Learning and Computer-Based Modeling (CCL) (Uri Wilensky, Director) for engineering the scoopers, creating the computer images, and 3D-print production.