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Applied Probability

via Brilliant


How can we accurately model the unpredictable world around us? How can we reason precisely about randomness? This course will guide you through the most important and enjoyable ideas in probability to help you cultivate a more quantitative worldview.

By the end of this course, you’ll master the fundamentals of probability, and you’ll apply them to a wide array of problems, from games and sports to economics and science.


  • Intro to Probability: Think probabilistically and explore the wide-reaching applications of probability.
    • Thinking Probabilistically: Use probability to avoid logical fallacies and quantify rare events.
    • Using Outcomes: Calculate probabilities as fractions of the total count of possible outcomes.
    • Applications: Explore applications of probability drawn from physics, meteorology, and tennis!
  • Probability Rules: Get the basics down.
    • Rule of Sum and Rule of Product: When do you add probabilities and when do you multiply them?
    • Inclusion-Exclusion: Use Venn diagrams to make deductions about probabilities.
    • The Rule of Complement: Learn to simplify problems by shifting your perspective to consider the probability an event does NOT happen.
    • Problem Solving: Put your probability knowledge to the test solving some real-world problems!
    • Managing Expectations: Make choices by comparing the costs and benefits to find the best and worst possible outcomes.
  • Conditional Probability: Update your assumptions with the information around you.
    • Defining Conditional Probability: Explore the mathematics that governs the probabilities of dependent events.
    • Applying Conditional Probability: Practice calculating conditional probabilities.
    • Bayes' Theorem: How is the probability of A given B related to the probability of B given A?
    • Misconceptions: Explore problems where misconceptions commonly arise, including the Monty Hall problem.
    • Casework: To understand a complex scenario, systematically split up the big problem into small cases.
    • Conditional Expectations: Analyze games that have multiple stages of events and rewards.
  • Probability Applications: Sports, economics, science, and more.
    • The Tennis Problem: How is a player's probability of winning a point related to the probability they win the whole game?
    • Probability in Science: Apply probability to solve problems in genealogy, cancer research, biology, and physics.
    • Probability in Economics: Use probability and expected value to forecast financial futures.
    • Probability in Quality Control: Extend what you know about probability to the understand the work potential of assembly lines.
  • Advanced Techniques: Push the frontiers of your knowledge with these probabilistic techniques.
    • Geometric Probability: Use geometric reasoning to understand probability distributions of games with infinitely many outcomes.
    • Bijections: Discover the surprising power of counting the same thing in two different ways!
    • Recursion: Harness the power of knowing a general pattern that relates the early and late values in a sequence.
    • Markov Chains: Learn what Markov chains are and use them to model the weather.
    • Generating Functions: Explore a clever way to use polynomials to efficiently calculate probabilities.


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