Chapter 7 · Simulations

Section 7.2Designing a Simulation Involving Probability

When you want to estimate the probability of a specific sequence of events — like having a boy first and a girl second — a simulation lets you model the process using random numbers. In this section, you will design and run a simulation to estimate the probability of a specific outcome, confirm your results with larger trials, and connect your findings to theoretical probability.

Summary

SimulationA model that imitates a real-world process using randomness to estimate the probability of a specific event or sequence of events.

ComponentThe smallest individual part of a simulation — for example, one child in a two-child family scenario.

TrialA full execution of the scenario from start to finish — for example, simulating the genders of two children produces one trial.

Relative frequencyThe proportion of trials in which the desired outcome occurred. As the number of trials increases, the relative frequency becomes a more reliable estimate of the true probability.

Theoretical probabilityThe exact probability calculated using rules and formulas rather than simulation. In a later chapter, you will learn to calculate these values directly — simulations build the intuition that makes those formulas meaningful.

Random number generatorA tool that produces unpredictable integers within a range you define, used to assign random outcomes to each component of a simulation. You can use the generator built into this page to run your simulation trials.

Simulations give us a hands-on, visual way to estimate probabilities and build intuition before we learn to calculate them theoretically.

The Real World Statistics Learning Progression

Each section follows the same learning progression: understand the concept, practice the skill, think like a statistician, then apply your learning to authentic data.

Learn This Section

Whether learning independently or teaching a class, the learning path below is designed to help students master the concepts in this section. Page references correspond to the Real World Statistics instructional system.

Read the sectionRead pages 220–222 to reinforce the concepts introduced in the lesson.

Practice what you've learnedComplete the practice questions on page 223. Check your work using the Answer Key, then review and correct any questions you missed before moving on.

Think like a statisticianExplore Enrichment: Role-Based Statistical Analysis Tasks and the Chapter Dataset to apply statistical thinking in real-world contexts.

Need Extra Support? If you'd like additional guidance while working through the practice questions, explore the Worked Solutions Manual for step-by-step solutions and explanations.

From Learners to Statisticians

Examples of authentic statistical work supported by this section include:

Simulating the probability that a quality-control inspector finds two consecutive defective items in a production line where 20% of items are defective, using random digits to assign outcomes and running 100 trials to estimate the likelihood (e.g., a manufacturing analyst evaluating inspection protocols).
Estimating the probability that a basketball player who makes 70% of free throws will hit three in a row, by assigning digits 1–7 to a made shot and 8–10 to a miss, then simulating 80 trials and computing the relative frequency (e.g., a sports analyst preparing a game-day report).
Modeling the probability that a family of four children includes exactly two boys followed by two girls, using a random number generator to assign outcomes and confirming the estimate with 100 trials before comparing to the theoretical result (e.g., a statistics student exploring combinatorics in context).

You may use these ideas to design your own investigations or explore the Real World Statistics System.

Enrichment: Role-Based Statistical Analysis Tasks

Enrichment tasks place students in the role of a statistician working on real research problems, where they evaluate data, justify decisions, and communicate findings clearly. Each section includes multiple enrichment tasks, including one with a model response that demonstrates statistical reasoning in context, showing how evidence is evaluated, decisions are justified, and conclusions are communicated. This serves as a bridge to working with authentic datasets and independent statistical investigation within each chapter.

Available in the Interactive eBook at the end of the section (page 223) and in the Appendix of the Print Book (page 401).

Teaching with Real World Statistics?

These structured learning paths — including Enrichment tasks and Chapter Datasets — can be adapted or copied directly into Canvas, Google Classroom, Schoology, and other learning management systems to simplify lesson planning and student assignments.

Part of the Real World Statistics instructional system — bringing together print, interactive learning, videos, datasets, enrichment, worked solutions, and teacher resources.