Simulation and Randomness

Concept

Introduce simulation as a way to answer a question by running an experiment many times instead of solving it analytically, which makes problems accessible long before the corresponding mathematics is. Explain pseudorandomness: the generator is deterministic given its seed, so results are reproducible on demand — a property that is essential for grading, debugging, and scientific reporting, not a limitation. Build Monte Carlo intuition through the estimate-as-average idea, and emphasize the point beginners most often miss: a simulated result is itself uncertain, it varies from run to run, and more trials shrink that variability in a predictable way. Discuss how to report a simulation honestly by stating the number of trials and the seed, and how the same machinery underlies resampling, sensitivity analysis, and probabilistic reasoning about real data.

Practice

Use Python’s random module and NumPy’s random generators for uniform draws, integer draws, sampling with and without replacement, and shuffling. Set a seed and demonstrate that two seeded runs match exactly while two unseeded runs do not. Build a simulation as a function of the number of trials, so the same code can be run at different scales, and structure it with a loop that records each trial’s outcome for later summary. Estimate a probability as a proportion of trials, then rerun at increasing trial counts and observe the estimate stabilizing. Summarize outcomes with the statistics and plots from the previous session, including a histogram of simulated results. In-class activity: students simulate a simple random process and compare outcomes across runs and seeds. Homework: implement a simulation study and explain the results.