Data 89: Mathematical and Graphical Foundations of Probability

Data Science Undergraduate Studies Logo

Course Description

How do we reason about data using mathematical models of random events? This course will complement the computational and conceptual toolkit students develop in Data C8. Students will build the mathematical and graphical problem-solving skills needed for upper-division courses in data science. The course will integrate and rehearse key proficiencies developed in prerequisite math classes while offering a focused introduction to specific multivariate concepts expected in following courses. It will fold this practice into a narrative motivated by the introduction to reasoning with uncertainty provided by Data C8, while providing a basic foundation in probability.

Offerings

  1. Spring 2026

Prerequisites

Math 51/1A, N1A, 16A, or 10A, completed with a grade of C- or better, or Pass. We recommend Math 51/1A. Data C8 or both Stat 20 and one of CompSci 61A or Data C88C, completed with a grade of C- or better, or Pass. We recommend Data C8.

Course Objectives

Data 89 will introduce students to analytic and graphical methods that will be further developed in the upper-division data science courses. The course will develop a focused selection of calculus topics, in single and multiple variables, as motivated by a probability context. The probability narrative is tailored to bridge students’ prerequisite inferential experience (Data C8 or Stat 20) to the probability competencies expected in their upper-division courses (equivalents to Data C100, Data C140, Data C102). The course will emphasize four key skills: computing chances, calculating expectations, finding critical points, and visualizing functions in univariate and multivariate contexts. The course will help students translate between analytical conclusions, graphical reasoning, and qualitative reasoning by highlighting the parametric dependency of answers to problem context. Practicing these links will help students sharpen their mathematical intuition.

Student Learning Outcomes

  • Name and apply the rudiments of probability (axioms and algebra rules)
  • Reason with multiple random variables (relate joint, marginal, and conditional distributions; define dependence and independence; measure association through covariance and correlation)
  • Identify, manipulate, and transform the distributions of random variables (identify distributions from generating processes, distinguish density from mass, compute the distribution of combinations and functions of random variables)
  • Define expectations and relate to concentration phenomena
  • Compute chances and expectations from distributions by summation and integration, with an emphasis on techniques of summation and integration (single and multiple on finite and infinite domains; close infinite series)
  • Visualize functions of single and multiple variables and understand parametric dependency (monotonicity, convexity, comparison of asymptotic rates, linearization, Taylor approximation, relate gradients to level sets)
  • Find conditional and unconditional maxima of functions of single and multiple variables (identify modes of distributions and M-estimators; use Lagrange multipliers)
  • Reason with and change coordinates (Cartesian coordinates, vector spaces, polar coordinates, and symmetries)