Modeling Pooled-Sample Testing, Part II

The context for this Modeling Pull-Out involves modeling pooled- sample testing for diseases or drug usage in order to reduce costs and/or conserve resources used in testing. This Pull-Out is Part II in a two-part series. In Part I students developed models for two-sample testing for COVID-19 based on data collected through simulation. In Part II, students develop models from theory, which will confirm the form of the models developed in Part I. The materials for this series of Modeling Pull-Outs are adapted from Testing 1, 2, 3, a unit in COMAP's secondary school core curriculum Mathematics: Modeling Our World (MMOW).

A Preparation Reading reviews material from Modeling Pooled-Sample Testing, Part I. It includes an "In the News" which discusses the situation where a high incidence of COVID-19 can lead to increased wait times for results when samples are pooled.

In Activity 1, Visualizing the Expected Number of Tests, students use probability to develop area models for the two-sample pooled strategy. Beyond the basic model representing the probabilities associated with pooling two samples from the same population, students draw an area diagram that could represent pooling two samples from different populations. In addition, they use area models to examine a situation where the tests are not 100% accurate.

In Activity 2, Generalizing the Area Models, students replace specific values for the probability that a randomly chosen person has COVID with a variable. This allows them to develop a quadratic model for the relationship between the probability and the expected number of tests per pooled sample. Students are then faced with the challenge of finding the best strategy for conducting pooled-sample testing when there are equal numbers of samples from two different populations.

In Activity 3, Pooling Three or More Samples, students create models for the expected number of tests per sample when three samples are pooled. They consider two options for when the pooled sample is positive. For Option 1, individuals from a positive pool are tested in a specific order as needed to determine who is positive. For Option 2, all individuals from a positive pool are individually tested. In analyzing Option 1, students return to the method used in Modeling Pooled-Sample Testing, Part I and determine a model based on simulated data. Then they develop a model based on probability theory and find the two models give similar results. In analyzing Option 2, students create a model based on probability theory. This model can easily be scaled up for use when more than three samples are pooled.

Mathematical Content

• Basic probability
Addition rule for disjoint events
Complement rule
Multiplication rule for independent events
Expected value
Tree diagrams
• Solving a quadratic equation
Using the quadratic formula
By finding the point where two graphs intersect
• Simulation

Materials needed
• Graph paper (Activity 1)
• Graphing calculators
• POOL3 program for TI-84 and/or access to computers/ Internet for Python version of POOL3 program with Jupyter Notebook