This is a large collection of practice problems and solutions on counting and probability. A few of the problems were written by our tutors but most of the material was created by instructors at various universities and colleges for their math and statistics courses. The content here takes the form of worksheets, homework, and quizzes, with solutions provided in all cases. We’ve also included some of the best relevant references and videos available online.
High school students taking a course covering counting and probability will likely find some of the problems on “Counting Basics”, “Permutations and Combinations” and “Basic Probability” to be the most relevant. The remaining problems are standard material in college discrete math and statistics courses and may be useful for advanced high school students. We have also included some of the best open source references and videos on counting and probability to be found online. Please write to us at info@tutorterrace.com if you have any questions or suggestions. Happy learning!
This concise open source Discrete Mathematics text by Oscar Levin covers the fundamentals of counting and combinatorics at an introductory college level.
This clearly written open source text by Pishro-Nik (UMass Amherst) covers the basics of probability theory, discrete and continuous probabilities, the central limit theorem and Bayesian inference among other topics.
This open source text by Grinstead and Snell is an introduction to probability. It covers discrete and continuous probabilities, conditional probability, the central limit theorem and random walks among other topics.
This series of videos from an excellent MIT course by Prof Tsitsiklis covers the basics of counting and combinatorics (Lecture 4), the fundamentals of probability theory (Lecture 1), Conditional probability and Bayes’ rule (Lecture 2) as well as Discrete and Continuous probability distributions (Lectures 5 through 10), among other topics.
This remarkable video from 3Blue1Brown explains Bayes’ theorem.
This short video from 3Blue1Brown has a quick proof of Bayes’ theorem.
Counting Basics (multiplication and addition principles, factorial)
Worksheet, Solutions (E. Hallman, UC Berkeley) || Worksheet, Solutions; Quiz, Solutions (R. Zhao, UC Berkeley) || Worksheet, Solutions (Profs Rao and Walrand, UC Berkeley ) || Homework, Solutions (A Nrusimha UC Berkeley) || Worksheet, Solutions (J. Rowan, UC Berkeley) || Tutor Terrace Worksheet and Solutions
Using Counting Principles (complimentarity, inclusion-exclusion, pigeonhole)
Worksheet 1, Solutions; Worksheet 2, Solutions; Quiz, Solutions (J. Rowan, UC Berkeley) || Worksheet 1, Solutions; Worksheet 2, Solutions: Quiz, Solutions (R Zhao, UC Berkeley)
Counting Permutations, Combinations and Partitions
Homework, Solutions (Prof Everlove, University of Michigan) || Worksheet, Solutions (Prof Hildebrand, University of Illinois at Urbana Champaign) || Worksheet 1, Solutions; Worksheet 2, Solutions (R. Zhao, UC Berkeley) || Worksheet 1, Solutions (J. Rowan, UC Berkeley) || Worksheet 1, Solutions, Worksheet 2, Solutions, Worksheet 3, Solutions (E. Hallman, UC Berkeley) || Worksheet and Solutions (Prof Blitzstein, Harvard) || Tutor Terrace Worksheet
Basic Probability
Worksheet and Solutions (Prof Blitzstein, Harvard) || Worksheet, Solutions (E. Hall, UC Berkeley) || Worksheet, Solution (J. Rowan, UC Berkeley) || Worksheet, Solution (R. Zhao, UC Berkeley) || Homework 1, Solution; Homework 2, Solution (Prof Everlove, University of Michigan) || Tutor Terrace Worksheet
Conditional Probability and Bayes theorem
Worksheet 1, Solutions; Quiz, Solutions, Worksheet 2, Solutions (R. Zhao UC Berkeley) || Worksheet 1, Solutions; Worksheet 2, Solutions (Profs Rao and Walrand, UC Berkeley) || Worksheet , Solutions; Worksheet 2, Solutions (J. Rowan UC Berkeley) || Homework, Solutions (Prof Everlove, University of Michigan) || Worksheet, Solutions (Prof Hildebrand, University of Illinois) || Worksheet and Solutions (Prof Blitzstein, Harvard) || Worksheet, Solutions (E. Hallman, UC Berkeley)
Continuous Probability
Homework, Solutions (Prof Everlove, University of Michigan) || Worksheet 1, Solutions; Worksheeet 2, Solutions; Worksheet 3, Solutions (R. Zhao UC Berkeley) || Worksheet 1, Solutions; Worksheet 2, Solutions, Worksheet 3 Solutions (J. Rowan UC Berkeley) || Worksheet, Solutions (Profs Rao and Walrand, UC Berkeley)
Challenging Probability Problems
20 Problems in Probablility with Solutions (Dr. Gravner, UC Davis) || 50 Challenging Problems in Probability with Solutions (Dr. Mosteller, late Chair of the Harvard Statistics Department).
