A Chance at Monte Carlo
15,000 simulated random points within a unit square.Copyright 2011 CaitlinJo, Wikimedia Commons {PD} http://en.wikipedia.org/wiki/File:Pi_30K.gif
At its core, the LEGO® MINDSTORMS® product provides a programmable microprocessor. Students use the EV3 processor to simulate an experiment involving thousands of uniformly random points placed within a unit square. Using the underlying geometry of the experimental model, as well as the geometric definition of the constant π (pi), students form an empirical ratio of areas to estimate a numerical value of π. Although typically used for numerical integration of irregular shapes, in this activity, students use a Monte Carlo simulation to estimate a common but rather complex analytical form—the numerical value of the most famous irrational number, π.
Monte Carlo methods are a class of simulation and sampling techniques that investigate models at randomly chosen points. These methods are widely used by engineers, scientists and mathematicians to perform numerical integration of complex functions of many variables, having no closed form. Monte Carlo is a classical technique in particle physics. More recent innovations include the ability to simultaneously track multiple objects in computer vision, and generally to track the state of observed "particles." Such progress has been made possible by recognizing the link between Monte Carlo integration and Bayesian estimation.
After this activity, students should be able to:
- (advanced) describe the link between frequentist probability (or, frequentism) and geometry
- quantify how the proportion of areas is reflected in outcome of the experiment
- distinguish between the geometric definition of π is and the numerical representation of π
- assess the quality of an estimate using percent error (or standard error for more advanced classes)
- confidently design a simulated experiment