In this lesson of this Probability Fundamentals Course, you'll learn how to solve complex probability problems, including concepts such as set complements, the multiplication rule, independent and non-independent events, and more.

Mathematicians are known for taking a complex problem and breaking it down into bits and pieces they can solve. Probability is no different! 

In this lesson, you'll use what you learned in the Estimating Probabilities and Probability Rules lessons to solve more involved probability problems. In addition, you’ll learn what it means for an event to occur with or without replacement. 

By the time you’ve completed this lesson, you’ll have learned skills that will allow you to answer the following probability questions:

  • What is the probability that it takes three flips or more for a coin to land heads up?
  • What is the probability of a coin landing heads up 18 times in a row?
  • What is the probability of getting at least one “six” in four throws of a single six-sided die?
  • What is the probability of getting at least one “double six” in 24 throws of two six-sided
  • What is the probability of getting four aces in a row when drawing cards from a standard 52-card deck?


  • Learn about set complements
  • Learn how to use complements to solve problems
  • Learn about the multiplication rule of probability
  • Learn about independent and non-independent events.

Lesson Outline

1. Compex Probability Problems
2. Opposite Events
3. Example Walk-Through
4. Set Complements
5. The Multiplication Rule
6. Independent Events
7. Combining Formulas
8. Sampling With(out) Replacement
9. Next Steps
10. Takeaways