Math Explorers (Grades 5-7)

Service Description

The Spring Semester includes 17 weeks of classes. There will be no class on (01/18, 02/15, 03/22, 04/05, and 05/24) Formal Probability & Statistics A Rigorous Course for Advanced Middle-School Students Course Format: Emphasis on definitions, reasoning, and problem solving Course Focus: This course presents probability and statistics as a formal mathematical discipline. Students learn to reason about uncertainty using precise definitions, logical structure, and carefully stated assumptions. The course builds a foundation for advanced study in mathematics, science, and quantitative fields. ⸻ Section I — Foundations of Probability Students begin by developing the formal language of probability. Experiments are described in terms of sample spaces and events, which are treated as sets. Fundamental set operations are introduced and used to combine and analyze events. The axioms of probability are then presented, establishing probability as a mathematical function governed by strict rules. Counting techniques, including permutations and combinations, are developed to systematically enumerate outcomes. These ideas culminate in the classical definition of probability for equally likely outcomes. Section II — Conditional Probability and Random Variables Building on the foundational framework, students examine how probabilities change when additional information is known. Conditional probability is introduced through restricted sample spaces, leading naturally to the formal concept of independence. The course then shifts from events to numerical descriptions of randomness through discrete random variables. Students study probability mass functions and learn how distributions encode probabilistic behavior. Expectation is introduced as a weighted average, providing a rigorous notion of “long-run average.” Section III — Variability, Data, and Statistical Reasoning In the final section, attention turns to variability and the connection between probability models and observed data. Students study variance and standard deviation as measures of spread, followed by empirical distributions derived from data. The informal Law of Large Numbers explains how theoretical probabilities emerge through repeated experiments. The course concludes with sampling and estimation, emphasizing bias, variability, and careful interpretation of results. Throughout, students are trained to state assumptions clearly and evaluate statistical claims critically.